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*    i  i 


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cop.l 


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of  the 

University  of  California 

Los  Angeles 

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cop.l 


This   book   is  DUE  on   the  last   date  stamped   below 


AUG  6      1925 
NOV  2  1  1932 


Form  L-9-5m-7.'oc< 


THE    WILEY    TECHNICAL    SERIES 

FOR 

VOCATIONAL    AND    INDUSTRIAL    SCHOOLS 


EDITED  BY 

JOSEPH    M.    JAMESON 

GIRARD  COLLEGE 


THE   WILEY   TECHNICAL   SERIES 

EDITED    BY 

JOSEPH  M.  JAMESON 

MATHEMATICS  TEXTS 

Mathematics  for  Machinists. 

By    K.    W.    Burnham,    MA        229   pages.     5  by   7. 
175  figures.      Cloth,  $1.75  net. 

Arithmetic  for  Carpenters  and  Builders. 

By  R.  Burdette  Dale,  M.E.     231  pages.     5  by  7. 
109  figures.     Cloth,  $1.75  net. 

Practical  Shop  Mechanics  and  Mathematics. 

By  James   F.   Johnson.      130  pages.     5   by   7.     81 
figures.      Cloth,  $1.40  net. 


CASS    TECHNICAL    HIGH    SCHOOL    SERIES 

Mathematics  for  Shop  and  Drawing  Students. 

By  H.  M.  Real  and  C.  J.  Leonard,     vii  +213  pages. 
4J  by  7.      188  figures      Cloth,  $1.60  net. 

Mathematics  for  Electrical  Students. 

Bv  H.  M.  Keal  and  C.  J.  Leonard,     vii  +  230  pages. 
41  by  7.      165  figures.      Cloth,  $1.60  net. 

Preparatory  Mathematics  for  Use  in  Technical  Schools. 
By  Harold  B.  Ray  and  Arnold  V.  Doub.  viii-)-68 
pages.     41  by  7.     70  figures.     Cloth,  $1.00  net. 


6-15-21 


MATHEMATICS 

FOR 

ELECTRICAL    STUDENTS 


BY 

H.  M.  KEAL 

Head  of  Department  of  Mathematics,  Cass  Technical 
High  School,  Detroit 


C.  J.  LEONARD 

Instructor  in  Mathematics,  Cass  Technical 
High  School,  Detroit 


46780 

NEW  YORK 

JOHN  WILEY  &  SONS,  Inc. 

London:  CHAPMAN  &  HALL,  Limited 
1921 


Copyright,  1921 

BT 

H.  M.  KEAL  and  C.  J.  LEONARD 


PREFACE 


The  purpose  of  these  two  brief  texts  is  to  give  to 
industrial  workers  and  students  who  have  not  com- 
pleted high  school,  that  part  of  algebra,  geometry,  and 
trigonometry  which  they  will  need  in  their  work.  The 
parts  of  algebra,  geometry,  and  trigonometry  necessary 
only  for  students  going  on  to  college  are  omitted.  The 
texts  are  planned  to  present  the  material  in  a  manner  that 
will  enable  the  student  to  master  it  with  a  minimum  of 
outside  help,  and  it  is  believed,  therefore,  that  they  are 
suitable  for  use  with  evening  school  and  continuation  classes, 
or  wherever  the  work  is  largely  individual. 

The  contents  of  the  course  may  briefly  be  outlined  as 
follows: 

1.  Equations. 

2.  Slide  Rule. 

3.  Formulas. 

4.  Positive  and  Negative  Numbers. 

5.  Proportion. 

6.  Quadratic  Equations. 

7.  Simultaneous  Equations. 

8.  Graphs. 

9.  Commonly  used  Geometric  Facts  (without  proof). 

10.  Logarithms. 

11.  Right  Triangles. 

12.  Oblique  Triangles. 


iv  PREFACE 

The  work  is  published  in  two  volumes;  one  especially 
adapted  to  those  interested  in  shopwork  and  drafting, 
the  other  to  those  interested  in  electrical  work.  The 
theoretical  parts  of  the  two  are  identical,  but  the  one  has 
applications  from  the  shop  and  the  other  from  electrical 
work. 

The  books  are  not  shop  or  electrical  texts  and  cannot 
be  expected  to  instruct  studento  along  shop  or  electrical 
lines.  They  are  mathematics  books,  written  for  shop  or 
electrical  students  and  workers,  with  applications  drawn 
from  the  shop  and  electrical  work.  The  main  purpose  of 
each  is  to  give  not  merely  a  knowledge  of  mathematics,  but 
to  furnish  that  understanding  of  mathematics  which  will 
increase  the  student's  practical  working  knowledge. 

The  chapter  on  geometry  is  intended  to  state  definitely 
the  geometrical  facts  that  most  students  know  in  a  more 
or  less  general  way.  All  of  the  geometry  ordinarily  needed 
for  trigonometry  and  its  uses  is  given,  and  the  less  evident 
theorems  are  illustrated  by  figures.  The  student  should 
study  the  entire  chapter  and  solve  the  problems  at  the  end 
as  a  test  of  his  understanding  of  geometrical  facts  and 
processes. 

The  chapter  on  the  slide  rule  is  not  a  discussion  of  the 
theory  of  the  rule  but  specific  directions  for  its  use. 

The  method  of  presentation  is  as  direct  and  definite  as 
possible,  so  that  the  student  can  study  the  text  by  himself, 
and  master  it  with  little  outside  help.  In  courses  where  in- 
struction on  the  slide  rule  is  given,  the  student  should  be 
urged  to  use  the  rule  as  much  as  possible,  especially  in  the 
solution  of  formulas  and  in  checking  logarithmic  problems. 

Many  of  the  self-evident  and  little  used  definitions  are 
omitted  from  the  text  for  the  sake  of  clearness.  A  dictionary 
of  terms  used  in  elementary  mathematics  is  given  in  the 


PREFACE  V 

appendix.  The  attention  of  students  should  be  called  to 
this  dictionary,  and  they  should  be  urged  to  use  it. 

Tables  of  formulas,  square  roots,  and  decimal  equiv- 
alents are  given  in  the  appendix  for  the  convenience  of  the 
student. 

Should  any  students,  after  completing  this  text,  desire 
to  fulfill  the  college  entrance  requirements  in  mathematics, 
they  will  find  that  the  knowledge  gained  from  this  course 
will  enable  them  to  cover  the  work  in  a  regular  college 
entrance  course  rapidly  and  efficiently,  or  the  following 
topics  added  to  the  work  of  this  course  will  meet  the  require- 
ments : 

Removal  of  Parentheses. 
Multiplication  (more  advanced  problems). 
Division  (Polynomial  by  Polynomial). 
Factoring. 

Fractional  Equations  (Polynomial  Denominator). 
Literal  Equations. 
Radicals. 

Equations  Containing  Radicals. 
Fractional,  Zero,  and  Negative  Exponents. 
Formal  Geometry. 

Relation  between  Trigonometric  Functions. 
Functions  of  the  Sum  and  Difference  of  two  Angles. 
Functions  of  twice  an  Angle  and  of  half  an  Angle. 
Proof  of  the  Law  of  Sines,  Law  of  Cosines,  Law  of 
Tangents. 

H.  M.  K. 

C.  J.    L. 
Detroit,  1921. 


TABLE  OF  CONTENTS 


CHAPTER  PAGE 

I.  The  Equation 1 

II.  The  Slide  Rule 13 

III.  Evaluation 29 

IV.  Positive  and  Negative  Numbers 49 

V.  Ratio  and  Proportion 68 

VI.  Cutting  Speed,  Pulleys  and  Gears 79 

VII.  Electrical  Formulas  Involving  Squares  and  Square 

Roots 86 

VIII.  Quadratic  Equations 95 

IX.  Simultaneous  Equations 104 

X.  The  Graph 109 

XI.  Geometry 119 

XII.  Logarithms 131 

XIII.  Right  Triangulation 148 

XIV.  Trigonometric  Functions  of  Any  Angle 164 

XV.  Oblique  Triangles 169 

XVI.  Electrical  Applications 177 

Appendix 

Dictionary  of  Terms 201 

Relation  between  Trigonometric  Functions 211 

Formulas 213 

Decimal  Equivalents 215 

Logarithms 216 

Logarithms  of  Functions.  .  . 218 

Natural  Functions 222 

vii 


MATHEMATICS 
FOR  ELECTRICAL  STUDENTS 


CHAPTER  I 


THE  EQUATION 

1.  Definition.  Scales  will  balance  only  when  equal 
weights  are  placed  in  the  pans  of  the  scales.  The  fact  that 
the  10-lb.  weight  in  one  pan,  Fig.  1,  is  exactly  balanced 
by  the  5-,  3-,  and  2-lb.  weights  in  the  other  pan  may  be 
expressed,. 

5+3+2=10. 


Fig.  1. 


Fig.  2. 


In  the  same  manner,  the  balance  of  the  scales  when  the 

unknown  weight  (x)  and  5  lbs.  are  placed  in  one  pan  and 

20  lbs.  in    the    other,  Fig.    2,    may  be    expressed    by  the 
equation, 

z+5  =  20. 

The   weight  of  (x)  may   be  found    by  removing  5  lbs. 

from  each  pan  of  the  scales,  leaving  (x)  in  one  pan  and 


2  THE   EQUATION 

15  lbs.  in  the  other  and   the  scales   still  in   balance.     This 
balance  may  be  expressed  by  the  equation, 

2=15. 

The  equation  x+5  =  20  is  a  statement  that  two  expres- 
sions are  equal.  The  expressions  2+5  and  20  are  called 
members  of  the  equation. 

To  solve  an  equation  is  to  find  the  value  of  the  unknown 
which  is  expressed  in  the  equation  by  a  letter. 

2.  Subtracting  from    Both  Members  of    an  Equation. 

Example .     Solve  x  +  5  =  20 

2+5  =  20 

2=15  (subtracting  5  from  both  members 
of  the  equation) 


EXERCISE   1 

Solve  for  the  unknown  in  each  equation: 

1. 

2. 
3. 
4. 
5. 

2+7  =12. 
2+9   =17. 

w+4i  =  8. 
2+2.5  =  5.5. 

Ans. 

5.        6.  2+3.3   =7.8.  Ans. 

8.        7.  a+  2\  =3|. 
\\.        8.  2  +  31|   =  42f. 
3f.        9.   2+1.25  =  2.35. 
3.        10.  6+1]      =2.35. 

4.5 

H. 

m 

i.i 

i.i 

-4^n^ 


17- 


Fig.  3. 


Equations  may  be  used  in  solv- 
ing other  problems  than  those  of  the 
balance. 

Example  1.  Let  x  =  the  missing 
dimension,  Fig.  3, 

2+4=17 

2=13 


CHECKING   AN   EQUATION 


Example  2.     What  number  added  to  14  equals  30? 
Let        x  =  the  number 
14+:r  =  30 
a;  =16 

EXERCISE  2 

Find  the  missing  dimensions,  using  equations: 
1  2  3 


.  *     n     . !  ^ 

K         J        >|< 

i- 

L_ 

r< 21 

_J 

32 


Fig.  5. 
Ans.   12. 

4.  A  vacant  lot  and  $192  were  exchanged  for  another 
lot  worth  $1000.     Find  the  value  of  the  first  lot. 

Ans. 
3.  Checking  an  Equation. 
In  the  equation  3a  +  5  =  32,     H  Fl  F1 
Fig.    7,   if    5    is    subtracted  |  —\ 

from  both  members,  the  equa-  A 

tion,  will  then  be,  Fig.  7. 

3a  =  27. 

The  value  of  the  unknown  (a)  may  then  be  found  by- 
dividing  both  members  by  3,  giving, 

a  =  9. 

The  problem  may  be  proved  by  substituting  9  in  place 
of  (a)  in  the  original  equation.     If  both  members  reduce 


THE   EQUATION 


to  the  same  number,  the  solution  is  correct.     A  complete 

solution  is, 

3a+5  =  32 

3a  =  27  (subtracting  5  from  both 
„  members) 
a  =  9     (dividing   both  members 
by  3) 
Check.  3X9+5  =  32 

27+5  =  32 
32  =  32 

EXERCISE   3 

Ans. 


Solve  and  check: 

1.  2a+12=   26.  7. 

2.  9y+15  =  96.  9. 

3.  12m  +  8  =  98.  7.5. 

4.  8+9z=116.  12. 

Find  the  missing  dimensions: 
9  10 


5.  28*+ 14   =128. 

6.  71m +.55  =  9. 07. 

7.  113+  1  =  \9-. 

8.  7wJ+5f  =  12f. 


Ans. 

.12. 
2. 
1. 


hJ" 


i                t 

u            -^ 

H — 

25 »i 

Fig.  8. 
Ans.  2.5. 


r\ 


L2 


A 


25 


5 


A 

Fig.  10. 


Fig.  9. 

Ans.  6. 
4.  Adding  to  Both  Mem- 
bers of  an  Equation.  The 
12-lb.  weight,  Fig.  10,  is  an  up- 
ward pull  on  the  pan  con- 
taining the  weight  (w).  If  the 
scales  balance  and  the  12-lb. 
weight  is    removed,  a    weight 


EQUATIONS   WITH   FRACTIONS  5 

of  12  lbs.  must  bo  placed  on  the  other  pan  to  balance  the 
scales.  Removing  the  upward  pulling  weight  is  the  same 
as  adding  the  weight  to  the  other  pan.  Thus,  expressed  as 
an  equation, 

w-12  =  25 

w  =  37  (adding  12  to  both  members) 
Example.        5x- 11  =  24 

53  =  35  (adding  11  to  both  members) 
x  =   7  (dividing  both  members  by  5) 
Check.  5X7-11  =  24 

35-11  =  24 
24  =  24 


EXERCISE   4 

Solve  and  check: 

1. 

2. 
3. 

•1. 

s-7  =  10.            Ans.  17. 
2x- 11  =  13.                  12. 
12s -34  =  26.                  5. 
2.1^-3.2  =  3.1.                3. 

5.  3z-9|  =  8.5.  Ans.  6. 

6.  17r-3|=13i           1. 

7.  43+11  =  17.               1.5 

8.  4x-ll  =  17.               7. 

9.  Find  the  unknown  weight 
in  Fig.  11,  if  the  scales  balance 
as  shown.  Ans.  lOf . 

10.  Three  times  a   number   [ 
less  26   is  equal   to   73.     Find 
the  number.  Ans.  33. 

5.  Equations  with  Fractions. 

Example.  £3  =10 


r~\ 


LO 


BBE 


a 


Fig.  11. 


Check. 


x  =  50  (multiplying  both  members 
by  the  least  common  de- 
nominator, 5) 
1X50=10 


THE   EQUATION 


From  this  example,  it  can  be  seen  that  both  members 
of  an  equation  may  be  multiplied  by  the  same  number. 
This  method  is  used  in  equations  where  fractions  occur. 
Example.        fa+£=3f. 

5z+6  =  51  (multiplying  both  members  by 
the  L.  C,  D.  15) 
5x  =  45  (subtracting  6  from  both  mem- 
bers) 
x=   9  (dividing  both  members  by  5) 


Check. 


3  X  "  ~rs  —  o-g 

Q2_Q2 

<  •  s  —  o  s 


EXERCISE   5 


Solve  and  check: 

Ans.     2. 


■!■•     2"£         3  —  3- 


2.  *x+34  =  9i. 


15. 


4.  1^-31  =  7. 

E       ,-M/         1  _  C2 


3.  ^—12  =  161^  means  ^/ 


Ans.  21. 
14. 

Ans.  84. 


6.  To  Solve  an  Equation.     Observe  that: 

I.  The  same  number  may  be  added  to  both  members. 

£  —  3  =  5 

x  =  8 

II.  The  same  number  may  be  subtracted  from  both  members. 

a:+7=12 
x—   5 

777.  Both  members  of  an  equation  may  be  multiplied  by  the 
same  number. 

re  =  14 


TRANSPOSITION  7 

IV.  Both  members  of  an  equation  may  be  divided  by  the 
same  number. 

3a:  =15 
x=   5 

Note.     In  x  —  5  =  8 

x  =  8+5  =  13 
and  z+7  =  12 

x  =  12-7  =  5 

The  effect  of  adding  or  subtracting  the  same  number  from  both  mem- 
bers of  the  equation  may  be  gained  by  transferring  (transposing)  the 
number  to  the  other  member  and  changing  the  sign  before  it. 

V.  An  equation  may  be  checked  by  substituting  in  the 
original  equation,  the  value  found  for  the  unknown.  If  both 
members  then  reduce  to  the  same  number,  the  solution  is  correct. 

VI.  In  an  equation,  such  as  2x-\-o  =  x-\-9,  2x,  5,  x,  and  9 
are  the  terms  of  the  equation,  2x+5  and  x+9  are  the  members 
of  the  equation  separated  by  the  sign  of  equality  ( = ) . 

Example  1.  A  man  saved  a  certain  amount  in  one 
year,  twice  as  much  in  the  second  year  and  three  times  as 
much  in  the  third  year  as  in  the  first.  In  all  he  saved 
$1200.     How  much  did  he  save  during  each  year? 

Let  x  =  amount  saved  first  year 

then  2x  =  amount  saved  second  year 

and  3a;  =  amount  saved  third  year 

z+2aH-3z=1200  (a;  means  la;) 

6a;  =  1200  (combining  like  terms) 
x=   200 
2x=   400 
3a;  =   600 


1200 


8  THE   EQUATION 

Example  2 .     x + 7x  -  3x  =  55 . 

5x  =  55  (combining  like  terms) 
x=ll 

Check.  11  +  77-33  =  55 

Example  3 .     7x  —  3x + x  =  55 . 

5x  —  55  (combining  like  terms) 
x=U 

Check.  77-33  +  11=55 

Example  4.     x—  3x+7:r  =  55. 

5x  =  55  (combining  like  terms) 
x=U 

Check.  11-33+77  =  55 

Note.  Observe  that  in  combining  x—Sx~t7x,  x  and  7x  were  first 
added  and  the  3x  then  subtracted.  In  combining  11—33  +  77,  11  and 
77  were  added  and  the  33  then  subtracted.  The  sign  before  each  term 
belongs  to  that  term.  (  +  )  is  understood  before  the  first  term  if  no 
sign  is  expressed. 


EXERCISE  6 

Solve  and  check: 

Ans. 

Ans. 

1. 

x+ll:c  =  24. 

2. 

4.  3z -  7x + lOz =36.        6. 

2. 

7m+3m=110. 

11. 

5.  x-ll.r+12.r  =  32.       16. 

3. 

17*-3W=45. 

3. 

6.  2x-  7+3z  =  28.            7. 

Solution. 

2x  - 

7  +  3x  =  28 

5x-7  =  28 

5x  =  35 

x=   7 

Check.  14-7  +  21=28 

Note.     Observe  that  the  7  cannot  be  combined  with  the  3.r  and 
2x  as  it  is  not  a  "like  term." 


UNKNOWN   IN   BOTH   MEMBERS  9 

7.  13m  -2m- 12  =  21.  Ans.     3. 

8.  4z  +  7  +  3x  =  91.  12. 

9.  &c-14z+9  +  7:c  =  37.  28. 

10.  $y-19-17y+12y=U.  10. 

11.  The  sum  of  three  numbers  is  32.  The  second  is 
twice  the  first  and  the  third  is  four  times  the  first.  Find 
the  numbers.  Ans.  4f,  9|,  18f. 

12.  A  man  saved  $800  in  three  years.  The  second  year 
he  saved  three  times  as  much  as  the  first,  and  the  third 
year  he  saved  $200.     How  much  did  he  save  the  first  year? 

Ans.  $150. 

13.  A  man  spent  $175  in  January.  He  paid  a  certain 
sum  for  room  rent,  twice  as  much  for  board,  as  much  for 
clothing  as  for  board,  and  $50  for  incidentals.  How  much 
did  he  spend  for  each?  Ans.  $25,  $50,  $50. 

14.  The  sum  of  \,  \,  and  |  of  a  number  is  equal  to  52. 
Find  the  number.  Ans.  48. 

7.  Equations  with  the  Unknown  Term  in  Both  Members. 
Example.        4x  =  2x+14. 

2x  =  14  (subtracting  2x  from  both  members) 
x  =   7 
Check.  28=14+14 

Example  2.     7x  =  48  -  bx. 

\2x  =  48  (adding  5x  to  both  members) 
x=   4 
Check.  28  =  48-20 

Example  3.    4m  —  32  =  2m. 

4m  =  2m +32  (adding  32  to  both  mem- 
bers 
2m  =  32  (subtracting    2ra    from    both 
members) 
m=  16 


10  THE   EQUATION 

Check.  64-32  =  32 

Note.     Observe    that    terms    containing    the    unknown    may    be 
added  to  or  subtracted  from  both  members. 


EXERCISE   7 

Solve 

and  check: 

1. 

3^-7  =  2/+5. 

Ans.  12. 

2. 

13m -42  =  6  +  w. 

4. 

3. 

2^+8+37/4-7  =  18+4?/. 

3. 

4. 

3x  —  4  =  x. 

2. 

5. 

3x-4=17. 

7. 

6. 

%x-2h  =  7|-2x  (clear  of  fractions).            3f . 

7. 

4  m  +  12  =  6m 

12  =  2m  (subtracting  4m  fr 

om  both  members) 

m  =    6 

Check.  24+12  =  36 

Note.     Observe  that  the  unknown  may  be  collected  on  either  side 
of  the  sign  of  equality.) 


8.  3?/+ 18=% +6.  Ans.  2. 

9.  2y-2$=5y-17l.  5. 

10.  17  =  2x-  3.  10. 

11.  ^+40  =  fn-7+4n.  14. 

12.  lx  +  62  =  3a;  +  2.  30. 

13.  |+2=y-lf  19. 

14.  7x  +  20-3x  =  60+4a:-50+8x.  \\. 

15.  3m  +  60=15m+3-2m  +  7.  5. 

8.  Equations  with  the  Unknown  in  the  Denominator. 
Equations  sometimes  occur  in  which  the  unknown  appears 
in  the  denominator.  In  such  equations,  the  unknown 
becomes  part  of  the  common  denominator. 


REVIEW 


11 


EXERCISE   8 


Solve  and  check: 
21  =  3 
2a    4' 


Ans.  14. 


42  =  3a      (multiplying     both     members     by     the 
L.  C.  D.  4a) 

39  +  10  =  7x    (multiplying    both   members  by  the 
L.  C.  b.Qx) 

3. 
12. 


3. 

11_ 

a 

=  3f. 

12 

4      2 

4. 

m 

m    3' 

5. 

15,25 
2s+4*       *' 

EXERCISE  9 

Review 

Solve 

and  check: 

1. 

2a; - 

-9  =  27. 

2. 

12z 

-9z  =  14- 

-Ax. 

3. 

15- 

-3aH-lLr  = 

=  39. 

8 

5      *      o, 

,  t 

A     —  —  —  —  -—  *}_L-I-- 

5     3     5     ^10^4- 
Write  equations  and  solve: 

5.  6. 


Ans.  18. 
2. 
3. 

6. 


^-3- 


7. 


kl25> 


u-3- 


*-2a>j 
-8Ji- 


-3?>|<.i,>l«.3Cfr 
.1G3 


-3.564- 


Fig.  12. 

Fig.  13. 

Fig.  14. 

Ans.  1|. 

Ans.   1.1. 

Ans.  2.908. 

12 


THE  EQUATION 


8.  Find  the  diameters  of  the  three  circles,  Fig.  15,  if  the 
diameter  of  the  second  is  twice  that  of  the  first  and  of 
the  third  H  that  of  the  first.  Ans.  3|,  Of,  5. 


Fig.  15. 


9.  Three  men  together  received  $335.  The  first  received 
$11  more  than  the  second  and  $16  less  than  the  third. 
Find  what  each  received.  Ans.  $99,  $110,  $120. 

10.  One  man  has  three  times  as  many  acres  of  land  as 
another.  After  selling;  GO  acres  to  the  second  man,  he  has 
still  40  acres  more  than  the  other.  How  many  acres  had 
each  at  first?  Ans-  80>  24()- 


CHAPTER   II 


SLIDE    RULE 


The  following  rules  and  directions  for  the  use  of  the 
slide  rule  are  not  intended  for  a  complete  manual  of  instruc- 
tions but  are  intended  to  present  in  a  simple  readable  manner 
instruction's  for  solving  the  problems  ordinarily  solved  on 
the  slide  rule.  Practice  problems  are  furnished  so  that  the 
student  will  have  had  some  practice  before  he  begins  to  use 
the  slide  rule  in  a  practical  way.  Theory  of  the  rule  will 
not  be  discussed,  but  considerable  attention  will  be  paid  to 
the  principles  of  operation  which  make  for  efficient  handling 
of  the  rule.  The  work  on  the  location  of  the  decimal  point 
may  be  omitted  if  desired. 

9.  Common  Types  of  Slide  Rules: 


BBB 


tlftLiiMJ^iffi 


i!.uu!JuJ...,.,.f..,.,.Li. 


UUu 


iA|i^^j44'^ 


^d^^^^^^iitf 


Fig.  16. — Mannheim  Slide  Rule.     Recommended  for  general  use. 


Fig.   17. — Polyphase  Slide  Rule.     Recommended  for  general   use  and 
in  some  problems  is  more  efficient  than  the  Mannheim  rule. 

13 


14 


SLIDE   RULE 


-rllwfrw^l'l'l'l'l'l'l'lllli 


mim< 


t» — 1[ — * 


-Uol — 


,  ■  t  ■  ,   iT'i  .1  /  ri.ri 


Z3Z 


Fig.  18. — Polyphase  Duplex  Slide  Rule.       Useful  in  problems  involv- 
ing ir  (=3.1416)  but  not  recommended  for  general  use. 

10.  Scales.  The  face  of  most  slide  rules  has  four  scales, 
the  A,  B,  C  and  D  scales.  The  A  scale  is  on  the  upper  part 
of  the  rule,  the  B  scale  on  the  upper  part  of  the  shde,  the  C 
scale  on  the  lower  part  of  the  slide  and  the  D  scale  on  the 
lower  part  of  the  rule.  The  polyphase  duplex  has  the  C 
and  D  scales  but  not  the  A  and  B,  the  other  two  types  have 
the  four  scales. 

The  C  and  D  scales  are  the  ones  in  most  common  use 
and  so  will  be  considered  first.  Notice  on  your  rule  that 
the  C  and  D  scales  are  alike.  Study  the  following  relations 
on  the  D  scale  by  reference  to  your  rule.  The  scale  as  a 
whole  is  divided  into  10  parts  marked  1,  2,  3,  4,  5,  6,  7, 
8  ,  9,  1,  as  in  Fig.  19. 


3  4 

Fig.  19. 


7     8     0    1 


These  divisions  correspond  to  the  numbers  1,  2,  3, 
4,  etc.,  but  the  numbers  on  the  slide  rule  have  no  decimal 
points,  so  the  3  stands  for  3,  30,  300,  3000,  etc.,  and  the 
4  for  4,  40,  400,  etc.  Numbers  between  1  and  2,  2  and  3, 
etc.,  must  be  read  on  the  unnumbered  divisions  between  the 
numbered   ones   or   sometimes   approximated   between   the 


SCALES  15 

small  division  lines.  Each  interval  1-2,  2-3,  3-4,  etc.,  is 
divided  into  10  main  subdivisions  indicated  by  the  longest 
lines  between  the  numbers.  The  main  subdivisions  between 
1  and  2  are  numbered  1,  2,  3,  —9;  but  they  are  not  so 
numbered  on  the  rest  of  the  rule.  Great  care  must  be 
taken  not  to  confuse  these  numbers  with  the  numbers  on 
the  main  part  of  the  scale.  Each  main  division  being 
divided  into  10  main  subdivisions,  these  main  subdivisions 
will  stand  for  11,  12,  13,  .  .  .  ,  21,  22,  23,  24,  .  .  .  ,31,  32, 
33,  ...  ,  41,  42,  etc. 

EXERCISE    1 

Locate  the  following  numbers  on  the  D  scale  and  have  the 
instructor  verify  three  or  four  of  your  answers: 

1.  5.  7.  55. 

2.  7.  8.  20. 

3.  65.  9.  19. 

4.  68.  10.  15. 

5.  23.  11.  18. 

6.  50.  12.  92. 

The  main  subdivisions  of  a  slide  rule  are  again  sub- 
divided into  ultimate  subdivisions.  Due  to  the  different 
lengths  of  the  divisions  some  of  the  main  subdivisions  are 
divided  into  10  spaces,  some  into  5  spaces,  and  others  into  2 
spaces.  These  ultimate  subdivisions  are  the  divisions  between 
23  and  24,  .  .  .  77  and  78,  etc.,  hence  they  correspond  to  the 
third  figure  of  a  number.  For  example,  find  15  and  16  on 
the  D  scale.  The  divisions  between  15  and  16  stand  for 
151,  152,  153,  154,  155,  156,  157,  158,  159. 


16  SLIDE   RULE 

EXERCISE  2 

Find  the  following  numbers  of  the  I)  scale  and  have  the 
instructor  verify  three  or  four  of  the  answers: 

1.  125.  6.  115. 

2.  146.  7.  105. 

3.  181.  8.  103. 

4.  195.  9.  108. 

5.  175.  10.  199. 

Between  2  and  4  the  main  subdivisions  are  all  divided 
into  only  five  ultimate  subdivisions,  hence  each  ultimate 
subdivision  stands  for  2  in  the  third  figure  of  a  number. 
For  example  find  22  and  23  on  the  D  scale.  The  divisions 
between  22  and  23  stand  for  222,  224,  226,  228. 

EXERCISE   3 

Find  the  following  numbers  on  the  D  scale  and  have 
the  instructor  verify  three  or  four  of  the  answers: 

1.  232.  6-  332. 

2.  254.  7.  368. 

3.  212.  8.  298. 

4.  210.  9.  108. 

5.  208.  10.  198. 

Between  4  and  the  end  of  the  rule,  the  main  subdivisions 
are  divided  into  only  two  ultimate  subdivisions,  hence  each 
ultimate  subdivision  stands  for  5  in  the  third  figure  of  a 
number. 


READING  BETWEEN  LINES  17 

EXERCISE  4 
Find  the  following  numbers  on  the  D  scale: 

1.  435.  6.  863. 

2.  565.  7.  995. 

3.  670.  8.  715. 

4.  675.  9.  710. 

5.  885.  10.  705, 

11.  Numbers  Not  Represented  by  a  Line.  Find  214  and 
216  on  the  D  scale;  then  215  is  midway  between  these  two 
marks.  To  find  2155,  set  the  cross  mark  of  the  runner  f 
of  the  way  between  214  and  216.  Find  420  and  425  on  the 
D  scale;  then  422  would  be  f  of  the  way  between  these 
two  marks.  Reading  the  numbers  between  the  lines  on  a 
slide  rule  is  not  exact  and  becomes  a  matter  of  judgment 
and  experience. 

EXERCISE   5 

Find  the  following  numbers  on  the  D  scale: 

1.  225.  9.  25. 

2.  365.  10.  125. 

3.  485.  11.  175. 

4.  477.  12.  1625. 

5.  585.  13.  115. 

6.  518.  14.  1125. 

7.  755.  15.  106. 

8.  757.  16.  1065. 


18 


SLIDE   RULE 


EXERCISE   6 
Read  the  number  represented  by  the  positions  of  Fig.  20. 


10         9 


\               I'l'l'l1!11!1!'!'!1         1       fl 

\                      4                  5              0            7 

k  8         9        1 

111        T 

12 

6 

Fig.  20. 


1. 

Ans.  23. 

2. 

27. 

3. 

17. 

4. 

145. 

5. 

216. 

6. 

77. 

9. 
10. 
11. 
12. 


Ans.  495. 
284. 
1165. 
103. 
422. 
715. 


12.  Division.  To  divide  6  by  3,  find  6  on  the  D  scale  and 
move  the  slide  to  the  right  until  the  3  on  the  C  scale  is 
directly  above  the  6.  Find  the  answer  on  the  D  scale 
under  the  end  of  the  C  scale.  Try  this  setting  on  your 
rule.  To  divide  20  by  5,  find  the  20  on  the  D  scale  and 
move  the  slide  until  the  5  on  the  C  scale  is  directly  above  the 
20  on  the  D  scale.  The  answer  is  on  the  D  scale  under  the 
end  of  the  C  scale. 

Rule.  To  divide  two  numbers  find  the  dividend  on  the 
D  scale  and  set  the  divisor  (on  the  C  scale)  directly  above  the 
dividend.  The  quotient  will  be  found  on  the  D  scale  under 
the  end  of  the  C  scale. 


MULTIPLICATION  19 

EXERCISE  7 

Divide: 

1.  6-3.  Ans.     2.       5.  168-5-2.  Ans.       84. 

2.  20 -=-4.  5.       6.  256X7.  36.6. 

3.  24 -=-8.  3.       7.  653-21.  31.1. 

4.  65-5.  13.       8.  768-125.  6.14. 

13.  Multiplication.  Multiplication,  in  arithmetic,  is  the 
opposite  of  division,  hence  the  operation  of  multiplication 
on  the  slide  rule  is  the  opposite  of  division  on  the  slide  rule. 
In  arithmetic  the  check  for  division  is  the  quotient  times  the 
divisor  should  equal  the  dividend.     Thus: 

6-4-3  =  2. 
Proof.  6  =  3X2. 

Therefore  to  multiply  2X3  on  the  slide  rule  the  2  will 
be  in  the  same  position  as  the  2  in  6—3;  that  is,  on  the 
D  scale  under  the  end  of  the  C  scale.  3  is  found  on  the 
C  scale  and  the  answer  on  the  D  scale  under  the  3.  Try 
this  on  your  rule.     Try  also  25X3. 

Rule.  To  multiply  two  numbers,  place  the  end  of  the 
C  scale  on  one  of  the  numbers  {on  the  D  scale)  and  find  the 
answer  on  the  D  scale  under  the  other  number  {on  the  C  scale). 

In  division  the  answer  was  under  one  end  of  the  C  scale 
in  some  problems  and  under  the  other  end  in  other  problems, 
and  so  in  multiplication  one  end  of  the  C  scale  is  used 
in  some  problems  and  the  other  end  in  other  problems. 
Use  the  end  which  will  bring  the  other  number  above  the 
D  scale.  For  example,  in  4X5  place  the  left  end  of  the 
C  scale  on  4  and  look  for  5  on  the  C  scale,  it  comes  beyond 
the  end  of  the  D  scale,  therefore  the  right  end  of  the  C  scale 
must  be  set  on  the  4  and  under  5  will  be  found  the  answer  20. 


SLIDE  RULE 

EXERCISE   8 

1. 

2. 
3. 
4. 
5. 

2X3. 
4X12. 

18X5. 

5X6. 

7X4. 

Ans. 

6.         6.  75X22. 
48.         7.  125X55. 
90.         8.  223X64. 
30.        9.  175X235. 
28.       10.  1575X45. 

20 


Ans.     1650. 

6775. 

14280. 

41100. 

70800. 

14.  Location  of  the  Decimal  Point.  Heretofore  the  loca- 
tion of  the  decimal  point  has  been  left  to  the  student  to 
determine  by  estimating  how  large  the  answer  should  be.  In 
practice  the  decimal  point  is  usually  located  in  this  manner, 
but  it  is  well  to  know  the  rules. 

Rule.  In  multi-plication,  if  the  slide  projects  to  the  left, 
add  the  number  of  places  in  both  factors  to  determine  the 
number  of  places  in  the  product.  If  the  slide  projects  to  the 
right,  add  the  number  of  places  in  both  factors  and  subtract  1 
to  determine  the  number  of  places  in  the  product.  In  division 
if  the  slide  projects  to  the  left,  subtract  the  number  of  places  in 
the  divisor  from  the  number  of  places  in  the  dividend  to  deter- 
mine the  number  of  places  in  the  quotient.  If  the  slide  projects 
to  the  right,  subtract  the  number  of  places  in  the  divisor  from 
the  number  of  places  in  the  dividend  and  add  1  to  determine 
the  number  of  places  in  the  quotient. 

The  following  diagram  should  prove  helpful  in  memo- 
rizing the  above  rules: 

Slide  projects  to  the 
Left  Righl 

Multiplication  Add  Add,  subtract  1 

Division  Subtracl  Subtract,  add  1 

In  the  above  rules  do  not  count  the  figures  in  the 
decimal  part  of  a  mixed  number.  In  the  case  of  decimals 
onlv  Hi.'  number  of  places  is  considered  negative  and  is 


CONTINUED   MULTIPLICATION  21 

equal  to  the  number  of  zeros  between  the  decimal  point  and 
the  first  significant  figure. 

EXERCISE   9 

Solve  the  following  problems  paying  attention  to  the 
location  of  the  decimal  point. 

1.  18-M2.  Ans.    1.5. 

2.  16X8.  128. 

3.  18-J-3.  6. 

4.  16X2.4.  38.4. 

5.  537X38.  20400. 

6.  1250X6.25.  7820. 

7.  2500X890.  2225000. 

8.  234X0.012.  2.808. 

9.  5620X3.45.  19400. 
10.  25-5-0.0062.  0.155. 

15.  Continued  Multiplication.  In  the  use  of  the  slide  rule 
great  care  must  be  taken  to  develop  efficiency  or  the  rule 
will  never  become  a  labor-saving  device.  In  finding  the 
product  of  3  factors,  as  12X14X24,  speed  and  accuracy 
can  be  gained  by  noting  that,  after  12  is  multiplied  by  14 
in  the  usual  way,  the  answer  appears  on  the  D  scale.  It 
will  not  be  necessary  to  read  this  number  as  it  is  in  place 
for  multiplying  it  by  24;  therefore,  mark  this  product  with 
the  crossmark  of  the  runner  and  then  multiply  the  number 
represented  by  this  position  by  24  by  drawing  the  end  of 
the  C  scale  under  the  crossmark  and  find  the  product  on 
the  D  scale  below  the  24  on  the  C  scale.  Try  this  on  your 
rule.  Marking  the  first  product  in  this  way  saves  time  and 
eliminates  the  error  of  reading  this  product  and  again 
setting  to  the  number  read. 


22 


SLIDE 

RULE 

EXERCISE    10 

1. 

2. 

3. 
4. 
5. 

2X3X4. 

12X7X13. 

250X3X78. 

172X305X450. 

751X0.046X231 

Ans. 

24. 

1092. 

58500. 

23600000. 

7980. 

Kl 
16.  Formulas  of   the   Type   R=~.     In  problems  of  this 

type  it  is  most  efficient  to  first  divide  K  by  A  and  then 

25X48 
multiply  by  /,     For  example,  R= — —r —      Divide  25  by 

32  in  the  usual  way;  the  result  appears  on  the  D  scale 
under  the  end  of  the  C  scale.  Then  to  multiply  this  number 
by  48,  find  48  on  the  C  scale  and  the  answer,  375,  appears 
on  the  D  scale  under  the  48  on  the  C  scale.  The  student 
should  keep  in  mind  the  fact  that,  in  problems  of  this  type, 
it  is  best  to  do  the  division  first,  since  this  leaves  the  rule 
set  for  multiplying  without  moving  the  slide.  All  problems 
of  this  type  can  be  done  with  one  setting  of  the  rule;  but 
in  some  cases  the  second  factor  will  appear  off  the  scale. 
It  is  then  necessary  to  change  ends  with  the  slide  before 
doing  the  multiplication.  This  is  accomplished  by  marking 
the  end  of  the  C  scale  with  the  crossmark  of  the  runner  and 
bringing  the  other  end  of  the  C  scale  under  the  runner. 
Changing  ends  of  the  slide  is  not  called  a  slide  rule  opera- 
tion. 

EXERCISE    11 

Ans.   15. 


17.5X32 
15 


37.3. 


PROPORTION  23 

225X13.2 
3-  "     790       *  376- 

25X3.3X21 
8X17 

17.  Proportion.  Set  on  your  rule  2  above  3,  and  notice 
that  any  other  pair  of  corresponding  numbers  are  in  the 
ratio  of  2  to  3.  For  example,  4  is  above  6  and  12  is  above 
18,  etc.  For  any  setting  of  the  slide  any  two  pairs  of 
corresponding  numbers  are  in  the  same  ratio.  This  makes 
a  simple  method  for  solving  proportion. 
For  example, 

16=_z 
27~64* 

Place  16  above  27  and  find  x  above  64. 
In  some  problems  it  will  be  necessary  to  change  ends 
with  the  slide  after  first  setting  for  the  one  ratio. 

Observe  that: 

When  in  any  problem  a  number  appears  off  scale,  it  is 
necessary  to  change  ends  with  the  slide. 

EXERCISE    12 

Ans.     6. 

30. 

85.8. 
80.8. 
11690. 


1. 

2     x 
3~9' 

2. 

3_18 

5      x ' 

125     371 

3. 

x   ~255' 

4. 

x        350 
289     1250' 

R 

24.5     7500 

24  SLIDE    RULE 

18.  The  A  and  B  Scales.  Most  slide  rules  have  an  A  scale 
on  the  upper  part  of  the  rule  and  a  B  scale  on  the  upper 
part  of  the  slide.  Make  sure  from  the  instructor  or  other- 
wise, that  your  rule  has  these  two  scales  before  proceeding 
with  the  following  directions: 

The  A  and  B  scales  are  the  same  and  are  composed 
each  of  two  slide  rule  scales  placed  end  on  end.  The  two 
scales  will  be  distinguished  as  the  right  scale  and  left  scale. 
Numbers  are  read  as  follows:  On  the  left  scale  1,  2,  3, 
4,  5,  6,  7,  8,  9,  10,  and  on  the  right  scale  10,  20,  30,  40, 
50,  60,  70,  80,  90,  100;  100,  200,  300,  etc.,  appear  again 
on  the  left  scale  and  1000,  2000,  3000,  appear  on  the  right 
scale.  Or,  in  other  words,  the  4  on  the  left  scale  stands 
for  4,  400,  40000,  etc.,  while  the  4  on  the  right  scale  stands 
for  40,  4000,  etc.,  and  not  for  4  or  400,  etc. 

Rule.  Numbers  having  a?i  odd  number  of  places  are  found 
on  the  left  scale,  and  numbers  having  an  even  number  of  places 
are  found  on  the  right  scale. 

In  counting  the  number  of  places  in  a  number  do  not 
count  the  figures  in  the  decimal  part  of  a  mixed  number. 
In  the  case  of  decimals  only,  the  number  of  places  is  deter- 
mined by  the  number  of  zeros  between  the  decimal  point  and 
the  first  significant  figure.  Thus  .04  is  found  on  the  left  scale 
and  .00423  is  found  on  the  right  scale.  A  decimal  with  no 
zeros  before  the  significant  figure  is  found  on  the  right  scale. 

The  A  and  B  scales  being  shorter  than  the  C  and  D 
scales  there  are  not  as  many  ultimate  subdivisions.  The 
student  should  study  the  scales  immediately  to  determine 
the  number  of  and  value  of  the  ultimate  subdivisions. 

19.  Use  of  he  A  and  B  Scales.  The  A  and  B  scales  are 
used  in  connection  with  the  C  and  D  scales  to  find  squares 
and  square  roots  and  to  solve  problems  involving  squares 
and  square  root. 


USE  OF  A   AND  B  SCALES  25 

Find  several  numbers  of  the  D  scale  as  2,  5,  12,  8,  34, 
and  notice  that  the  squares  of  these  numbers  4,  25,  144, 
64,  1156,  appear  on  the  A  scale  directly  above  the  numbers 
on  the  D  scale.  The  same  relation  exists  between  the 
B  scale  and  C  scale.  To  find  the  square  root  of  a  number, 
find  the  number  on  the  A  or  B  scale,  and  find  the  square 
root  on  the  D  or  C  scale  directly  below  the  number. 

EXERCISE    13 
Find  the  square  root  of  the  following  numbers: 

1.  4.  Ans.     2.       4.     563.  Ans.  23.7. 

2.  36.  6.       5.     1582.  39.8. 

3.  144.  12.       6.  75000.  274. 

In  handling  formulas  involving  squares  and  square 
roots  do  the  multiplications  and  divisions  on  the  A  and 
B  scales,  using  the  C  and  D  scales  only  to  locate  such 
numbers  as  172,  3522,  etc.,  or  to  find  the  square  root  of  a 
product  or  quotient.  The  rules  for  multiplication  and 
division  on  the  A  and  B  scales  are  the  same  as  for  multi- 
plication and  division  on  the  C  and  D  scales. 

Example  1.  7X52=?  Find  7  on  the  A  scale  and 
set  the  right  end  of  the  B  scale  under  it  the  answer  will 
appear  on  the  A  scale  above  52  on  the  B  scale,  that  is, 
above  the  5  on  the  C  scale ;  use  the  crossmark  on  the  runner. 
It  is  not  necessary  to  read  the  value  of  52. 

Example  2.     -7^=?     Find  36  on  the  A  scale  place  22 

on  the  B  scale  under  it,  using  the  2  on  the  C  scale  to  locate 

the  22  on  the  B  scale.     The  answer  is  on  the  A  scale  above 

the  end  of  the  B  scale. 

122 
Example  3.     -75-  =  ?     Find  122  on  the  A  scale  (above 
18 


26  SLIDE    RULE 

12  on  the  D  scale);   set  18  on  the  B  scale  below  122.     The 
answer  appears  on  the  A  scale. 

Example  4.  Vff=?  Or  V23X7  =  ?  Do  the  multi- 
plication or  division  on  the  A  and  B  scales  and  the  answer 
will  be  found  on  the  C  or  D  scale  below  the  result  of 
multiplication  or  division  on  the  B  or  A  scale.  It  is  not 
necessary  to  read  the  result  on  the  A  or  B  scale. 

Example  5.       — -^ — =  ?     This     is     similar     to     the    -j- 
o  A 

formula,   and   so   12   should   be   divided   by   52   first,   as  in 

Example   2.     This  leaves  the  rule   set  for  multiplying  by 

18  without  resetting  the  slide. 

20.  General  Suggestions.  The  following  suggestions 
should  prove  helpful  in  the  use  of  the  slide  rule. 

1.  The  A  and  B  scales  are  used  in  square  and  square  root 
formulas,  the  C  and  D  scales  being  brought  into  use  to  locate 
such  numbers  as  272,  etc.,  or  to  find  a  square  root. 

2.  The  principles  of  operation  of  the  A  and  B  scales  are 
the  same  as  for  the  C  and  D  scales. 

3.  In  division  the  dividend  may  be  located  on  either  scale 
and  the  divisor  on  the  opposite  scale,  but  the  quotient  will 
always  appear  on  the  scale  with  the  dividend. 

4.  In  formulas  involving  more  than  one  operation  keep  in 
mind  where  the  result  would  appear  after  each  operation. 
It  is  riot  necessary  to  read  these  intermediate  results. 

5.  Division  leaves  the  rule  set  for  multiplication  without 
moving  the  slide. 

21.  Areas  of  Circles.  Finding  the  area  of  circles  is  similar 
to  Example  1  above,  but  it  will  be  noticed  that  -k  (  =  3. 1416) 
is  marked  with  a  special  mark  on  the  A  and  B  scales.  To 
find  the  areas  of  a  circle  place  the  end  of  the  B  scale  on  t 
on  the  A  scale  and  find  the  areas  on  the  A  scale  above  the 
radius  square  of  the  B  scale  (that  is,  above  the  radius  on 


READING  FROM  ONE  SETTING  27 

the  C  scale).  With  this  one  setting  the  areas  of  any  number 
of  circles  can  be  read  off  on  the  A  scale  above  the  radius  on 
the  C  scale. 

22.  Ratios  of  the  Same  Kind.  Different  values  can  be 
read  from  one  setting  of  the  rule  in  many  other  problems. 
For  example,  one  setting  makes  it  possible  to  read  off  the 
number  of  centimeters  corresponding  to  any  number  of 
inches,  or  to  read  the  number  of  pounds  corresponding  to 
any  number  of  kilograms,  or  many  other  ratios  based  on 
the  same  constant.  A  table  of  these  various  settings  and 
proportions  will  be  found  on  the  back  of  most  single  face 
rules  and  in  the  slide  rule  manual. 


1.  27X83. 

2.  6X17X35. 

3.  167-5-14. 

.  27X35 


EXERCISE 

14 

Review 

Ans. 

2241. 
3570. 

11. 

93 

13     * 

5. 

16X83X41 

125X7 

6. 

212. 

7. 

Vl34. 

8. 

322X5. 

9. 

18X172. 

10. 
11. 

(9X64)2. 
V35X72. 

12. 

/l39 

\  22"' 

13 

27X22 

172 


72.7. 

62.2. 

441. 

11.58. 
5120. 
5200. 
332000. 
50.2. 

2.514. 
2.06. 


28  SLIDE  RULE 


14.  ??.  71.3. 


16.  V 24X169.  63.7. 

17.  Find    the    areas    of    the    circles   having    for   radius 
(1  setting): 

(a)     3  in.  Ans.       28.3. 

(6)   14  in.  616. 

(c)  37  in.  4300. 

(d)  2.68  in.  22.6. 

18.  Change  to  centimeters  the  following  inches  (1  setting) : 

(a)  8.  Ans.     20.3. 

(b)  67.  170. 

(c)  12.3.  31.2. 

(d)  2.26.  5.74. 

19.  Change  to  inches  the  following  centimeters: 

(a)     27.  Ans.      10.63. 

(6)      19.  7.48. 

(c)  184.  72.4. 

(d)  643.  253.1. 

The  work  of  this  chapter  is  not  a  complete  slide  rule 
manual,  but  the  student  should  be  able  to  do  the  problems 
ordinarily  done  on  the  slide  rule  rapidly  and  efficiently. 
He  needs  still  more  practice  to  become  accurate,  rapid, 
and  confident.  There  is  still  much  to  be  learned  on  the 
slide  rule,  and  the  student  should  begin  to  study  the  manual 
furnished  with  slide  rules,  and  to  watch  his  own  work  to  see 
where  he  can  increase  his  speed  and  efficiency. 


CHAPTER  III 
EVALUATION 

23.  General  Numbers.  The  area  of  a  rectangle  is 
found  by  multiplying  the  base  by  the  altitude.  This  may 
be  expressed  by  (b)X(a),  in  which  the  value  of  (6)  may  be 
12  ft.,  7  in.,  25  rods,  or  any  number  of  any  unit  used  to 
measure  length,  and  (a)  may  be  any  number  of  a  like  unit. 
Letters  which  may  represent  different  values  in  different 
problems  are  called  general  numbers.  The  unknown  letters 
used  in  Chapter  I  are  general  numbers. 

EXERCISE    1 

1.  Find    b+a    when  6  =  3,     a  =   7.    Ans.  10. 

2.  Find    b+a    when  6  =  5,     a  =12.  17. 

3.  Find  —  -\ —  when  m  =  3,     n  =  4, 

n     y 

x  =  5,     y  =  8.  If. 

n 

4.  Find  R  —  -j  when  #  =  5,     n  =  4, 

d=8.  4§. 

24.  Signs.  When  the  multiplication  of  two  or  more 
factors  is  to  be  indicated,  the  sign  of  multiplication  is  often 
omitted  or  expressed  by  the  sign  (•);  thus  7XaXbXm  is 
written  7-a-b-m,  or  more  often  7a6m.  Care  must  be  taken 
in  the  use  of  the  sign  ( • )  to  distinguish  it  from  the  decimal 
point,  7-9  means  7X9,  while  7.9  means  7^. 

29 


30  EVALUATION 

EXERCISE  2 

1.  Find  ax  when  a  =  3,  x  =  5.  Ans.  15. 

2.  Find  3mn  when  m  =  2,  n  =  7.  42. 

Note.  When  addition,  subtraction,  multiplication  and  division 
occur  in  the  same  problem,  do  the  multiplication  and  division  first  in 
the  order  in  which  they  occur,  then  do  the  addition  and  subtraction. 

3.  Find  ax+by  when  a  =  4,  x  =  7, 

6  =  3,  ^  =  4.      Ans.  40. 

4.  Find  2nd—  3cd  when  a=l2,  d  =  4, 

c=5.  36. 

25.  Coefficient.  When  the  multiplication  of  two  factors 
is  expressed  as  ax,  either  factor  is  called  the  coefficient  of 
the  other. 

EXERCISE   3 

What  is  the  coefficient  of  x  in  the  following: 

1.  bx. 

2.  5x. 

3.  3ax. 

4.  (a+b)x. 

5.  (2d-S)x. 

6.  xy. 

7.  x(m  —  n). 

26.  Power.  If  all  the  factors  of  a  product  are  the 
same,  the  product  is  called  a  'power  of  that  factor.  Thus 
xxxx  is  x  fourth  power  and  is  written  x4.     Similarly, 

xx       =x2,  and  is  read  x  2d  power  or  x  square. 
xxx     =z3,  and  is  read  x  3d  power  or  x  cube. 
xxxx   =x4,  and  is  read  x  4th  power. 
xxxxx  =  x5,  and  is  read  x  5th  power. 


NOTATION  31 

27.  Exponent.  The  number  which  indicates  the  power 
is  written  as  a  small  number  above  and  to  the  right  of  the 
factor  and  is  called  its  exponent.  For  example,  in  x2,  2 
is  the  exponent,  in  y3,  3  is  the  exponent. 

28.  Base.  The  number  wh  ch  is  used  as  a  factor  a 
number  of  times  is  called  the  base.  For  example,  in  r3, 
x  is  the  base  and  3  is  the  exponent;  in  53,  5  is  the  base, 
3  is  the  exponent  and  the  power  is  equal  to  125. 

EXERCISE  4 

1.  Find  x2          when  x  =  5.  Ans.     25. 

2.  Find  m3         when  m=2.  8. 

3.  Find  xi  —  y2  when  x  =  3,  y  =  5.  56. 

4.  Find  orb3       when  a  =  4,  6  =  2.  128. 

5.  Find  ax3        when  a  =  6,  x  =  4.  384. 

6.  Find  5x3        when  x  =  4.  320. 

29.  Signs  of  Grouping.  The  sign  of  grouping  most 
commonly  used  is  the  parenthesis  (  ).  It  means  that  the 
parts  enclosed  are  to  be  taken  as  a  single  quantity.  For 
example:  3(x+y)  means  3  times  the  sum  of  x  and  y;  (x+y)3 
means  (x+y)(x+y)(x+y). 

EXERCISE  5 

1.  Find  3(15-7).  Ans.  24. 

2.  Find  3 (x +y)  when  x  =  2,  y  =  4.  18. 

3.  Find  2 (x2-  a3)  when  x  =  7,  o  =  3.  44. 

4.  Find  5(2x+3ay)       when  x  =  4,  a  =  2,  y  =  6.       220. 

5.  Find  bx(3m2 - 2n2)   when  x  =  2,  m  =  8,  n= 5.     1420. 

30.  Evaluation.  Evaluation  of  an  expression  is  the 
process  of  finding  its  value  by  substituting  definite  numbers 
(figures)  for  general  numbers  and  performing  the  operations 
indicated. 


32  EVALUATION 

EXERCISE  6 

Evaluate: 

1.  'Sa2x:i  when  a =3,  x  =  2. 
Solution.  3  a2  z3 

3-32-23 
3-9-8 
216.       Ans. 

2.  2z(a3  —  ?/)  when  z  =  5,  a  =  4,  ?/  =  30. 

Solution.  2x(a3  —  y) 

2-5(43-30) 

10(64-30) 

10-34 

340.       Ans. 

3.  2:r  when  a;  =  5.  Ans.     10. 

4.  x2    when  rr  =  5.  25. 

5.  30+?/)     when  s  =  2,  y  =  4.  18. 

6.  4(a+6)2   when  a  =  2,  6=4.  144. 

7.  x2y3  when  a- =  3,  ?/  =  2.  72. 

8.  7x2  when  x  =  4.  112. 

9.  x3  —  a2       when  z  =  5,  a  =  3.  116. 

10.  2a%3(a-6)  when  a=12,  x=2,  6  =  8.  9216. 

11.  1+-+|  when  2  =  2,  n  =  8, 

31.  Formulas.  A  formula  is  the  statement  of  a  rule 
or  principle  in  terms  of  general  numbers.  For  example, 
the  area  of  a  rectangle  is  equal  to  the  product  of  the  base 
by  the  altitude.     Stated  as  a  formula  this  becomes    A=ba. 

32.  Evaluation  of  Formulas.  Evaluation  of  a  formula 
is  the  process  of  substituting  definite  numbers  for  general 


FORMULAS  33 

numbers,  and  solving  the   resulting   equation    for  the  one 
remaining  general  number.     For  example,  evaluate: 

A=%bh  when  6  =  5,  h  =  12. 

Substituting  values, 

A  =  |-5-12 

Whence, 

A=30 

Evaluate, 

x-\-3r  =  5a  when  r  =  2,  o  =  7 

Substituting  values, 

.r+3-2  =  5-7 

Multiplying, 

.r+6  =  35 
Whence, 

.r  =  29 

33.  Electrical    Formulas — Resistances    in    Series.     The 

total,  or  combined  resistance,  R,  of  the  three  resistances  R\, 


Rj  R2  Rs 

Fig.  21. 

R%  and  Rs  in  series,  Fig.  21,  is  expressed  by  the  equation 

R  =  Rl+R2  +  R3. 
All  are  measured  in  ohms. 


34 


EVALUATION 

1 

EXERCISE  7 

£va 

Urate  1 

the  formula 

,  to  find  the 

missing 

values 

R 

Ri 

R<2 

R?. 

1. 

12 

5 

8 

Ans. 

25. 

2. 

G.3 

2.9 

4.43 

13.63 

3. 

6.8 

4.6 

2.01 

13.41 

4. 

10 

2 

1 

7. 

5. 

120 

6 

70 

44. 

6. 

16 

3 

4 

9. 

7. 

25 

12.5 

2.6 

9.9. 

8.  19.1         14  4.41  0.69. 

9.  20  6.4         9.98  3.62. 

10.  The  combined  resistance  of  four  lamps  in  series  is 
1025  ohms.  The  first  lamp  has  a  resistance  of  250  ohms, 
the  second,  260  ohms  and  the  third  290  ohms.  What  is 
the  resistance  of  the  fourth  lamp?  Ans.  225. 

11.  Find  the  total  resistance  of  three  coils  in  series  if 
the  first  has  a  resistance  of  2  ohms,  the  second  5  ohms,  and 
the  third  6  ohms.  Ans.   13. 

12.  Four  storage  batteries  are  connected  in  series  for 
charging,  3  have  internal  resistances  of  4  ohms  each  and  the 
fourth  has  an  internal  resistance  of  6  ohms.  How  many 
ohms  must  be  put  in  series  to  make  a  total  resistance  of 
25  ohms?  Ans.  7. 

34.  Ohm's  Law.  Ohm's  law  for  the  relation  of  voltage 
to  current  and  resistance  in  a  circuit  is  expressed  by  the 

formula: 

E  =  IR 
where,     E  =  voltage, 

7  =  current  measured  in  amperes, 

R  =  resistance  measured  in  ohms. 


ELECTRIC  CURRENT  35 

EXERCISE  8 
Evaluate  the  formula  to  find  the  missing  factors: 


E 

I 

R 

1. 

.55 

11 

Ans. 

6.05. 

2. 

.75 

9 

6.75. 

3. 

110 

220 

.5. 

4. 

110 

.75 

146.7. 

5. 

220 

450 

.488 

6. 

6 

1.5 

4. 

7.  If  7=§3  find  7  when  E  =110  volts  and  /?  =  9ohms. 

K  Ans.   12.2. 

8.  If  R-j,  find  R  when  £=110  volts  and  7  =  0.5 
amperes.  Ans.  220. 

9.  How  much  current  will  flow  through  the  windings 
of  an  electromagnet  of  140  ohms  resistance,  when  placed 
across  a  110  volt  circuit?  Ans.    .185. 

10.  What  is  the  resistance  of  an  incandescent  lamp  on 
a  110  volt  circuit  if  the  current  is  1.5  amperes?        Ans.  73.3. 

11.  Four  lamps  in  series  have  resistances  of  2  ohms, 
3  ohms,  2.5  ohms  and  3.25  ohms.  If  119  volts  are  applied 
find  the  number  of  amperes  current  flowing.    Ans.   1 1 .  07. 

12.  An  electric  car  heater  is  supplied  with  500  volts 
from  the  trolley.  The  current  is  2.5  amperes,  what  is  the 
resistance?  Ans.  200. 

13.  Find  the  current  sent  through  a  circuit  of  200  ohms 
resistance  by  a  Daniell  cell  cell  which  has  a  voltage  of  1 .  09 
volts.  Ans.  0.00545. 

35.  Work  Done  by  an  Electric  Current.  The  work  done 
by  an  electric  current  equals  the  power  times  the  time. 
Expressed  as  an  equation  this  is  W  =  PT.  The  work  is 
measured  in  watt-hours  or  kilowatt-hours,  the  power  in 
watts  or  kilowatts,  and  the  time  in  hours. 


36  EVALUATION 

EXERCISE  9 

Evaluate  the  formula  to  find  the  missing  factors: 

W  P  T 

1.  110  55  Ans.       2. 

2.  75  7  525. 

3.  485  3.25  149.23. 

4.  What  will  be  the  cost  at  4  cents  per  kilowatt-hour 
to  run  a  4400  watt  heater  for  (i  hours?  (1  kilowatt  =  1000 
watts.)  Ans.  $1.00. 

5.  How  much  work  is  done  when  a  40  watt  lamp  is 
lighted  for  4  hours?  Ans.   160  W.H. 

6.  The  power  used  in  doing  a  piece  of  work  is  100  kilo- 
watts and  the  time  taken  is  3  hours.  How  much  work  is 
done?  Ans.  300  K.W.H. 

7.  The  work  done  by  a  heater  is  equal  to  50  kilowatt- 
hours,  and  the  time  is  200  min.     What  is  the  power? 

Ans.  20  K.W. 

8.  The  power  used  by  an  arc  lamp  is  500  kilowatts  and 
the  work  done  is  2500  kilowatt-hours.     Find  the  time. 

Ans.  5  Hrs. 

36.  Heat  in  an  Electric  Circuit.  The  heat  developed  by 
an  electric  current  is  expressed  by  the  formula: 

H  =  0. 24  Pt, 

where,    11  =  Heat  in  calories,* 
P  =  power  in  watts, 
t  =  time  in  seconds. 

The  constant  0.24  is  called  the  heat  equivalent  of  elec- 
tricity. 

*  A  calorie  is  the  amount  of  heat  required  to  raise  the  temperature 
of  one  gram  of  water  one  degree  Centigrade. 


TRANSFORMATION  OF  FORMULAS  37 

EXERCISE   10 

Evaluate  the  formula  to  find  the  missing  factors: 

H  P                     t 

1.  89  1.5  hrs.  Ans.   115344. 

2    230000  3  hrs.  20  min.  79 . 8. 

3.  189000  67                                                     3.26. 

4.  How  much  heat  is  generated  per  hour  in  an  electric 
iron  using  660  watts?  Ans.  570240. 

37.  Electricity  Stored  in  a  Condenser.  The  electricity 
stored  in  a  condenser  is  expressed  by  the  equation : 

Q  =  CE, 

where,    Q  =  quantity  of  electricity  measured  in  coulumbs, 
C  =  capacity  of  the  condenser  in  farads, 
E  =  applied  voltage. 

EXERCISE   11 

1.  45  volts  are  applied  to  a  condenser  of  2  microfarads 
capacity.  How  much  electricity  is  stored?  (1  microfarad 
=  .000001  farad.)  Ans.  0.00009. 

2.  How  much  voltage  is  required  to  store  0 .  005  coulumb 
in  a  condenser  of  250  microfarads  capacity?  Ans.  20. 

38.  Transformation  of  Formulas.  Ohm's  law  may  be 
expressed  in  any  one  of  the  following  forms: 

1.  E  =  IR. 

2.  R=f. 

1  is  the  most   convenient  for  finding  E,  2  for  finding 

46780 


38  EVALUATION 

R,  and  3  for  finding  I.  It  is  not  necessary  to  memorize 
more  than  one  of  these  formulas.  The  other  two  may  be 
derived  from  any  one  as  follows: 

1.  From  1  derive  2: 

E  =  IR 

V  J  D 

d=-d-  (Dividing  both  members  by  R) 
R      K 

E 

n  =  I     (Cancellation) 

2.  From  2  derive  1 : 

EI 

IR  =  -j-  (Multiplying  both  members  by  /) 

IR  =  E     (Cancellation) 

EXERCISE   12 

1.  From  1  derive  3. 

2.  From  Q  =  CE  derive  formulas  for  C  and  E. 

3.  From  Q  =  IT  derive  formulas  for  /  and  T. 

General  Directions.  To  change  the  form  of  a  formula 
first  clear  the  formula  of  fractions.  Then  get  the  letter 
to  be  solved  for  on  one  side  of  the  equation,  and  all  other 
letters  on  the  other  side. 

39.  Force  of  Attraction  or  Repulsion  on  a  Magnet.  A 
magnet  brought  into  a  magnetic  field  is  acted  upon  by  a 
force  expressed  by  the  formula: 

F=MH, 

where,     F  =  force  of  attraction  or  repulsion  in  dynes, 
M  =  strength  of  the  magnet  in  unit  poles, 
H  =  intensity  of  magnetic  field  in  gausses. 


RESISTANCES  IN  PARALLEL  39 


E 

XERCISE   13 

Evaluate  the  formula  to  find  the  missing  terms. 

F 

M 

H 

1. 

600 

3000             Ans.   1800000. 

2.     4000000 

2500                            1600. 

3.     65000 

200 

325. 

4.  A  magnet  of  600  unit  poles  is  placed  in  a  magnetic 
field  of  5000  gausses  intensity.  What  force  is  exerted  upon 
the  magnet?  Ans.  300000  dynes. 

5.  Solve  the  formula  F  =  MH  for  M  and  for  H. 


40.  Resistances  in  Parallel.     The  formula 

p    A  p    A  j-> 
.ttl       /V2       113 

is  used  for  finding  the  combined  or  equivalent  resistance 
of  resistances  connected  in  parallel  as  in  Fig.  22.  Ri,  R2, 
and  R3  are  the  separate  resistances,  R  is  the  combined 
resistance.     All  resistances  are  measured  in  ohms. 


40  EVALUATION 

EXERCISE   14 

Evaluate  the  formula  to  find  the  missing  terms. 

R  /?i  R2  Rs 

1.  15  8  12  Ans.  3.(34. 

2.  220  3  12  2.37. 

3.  210  210  210  70. 

4.  Three  lamps  of  195  ohms  each  are  connected  in  paral- 
lel.    Find  the  resistance.  Ans.  05. 

5.  Five  lamps  are  connected  in  series,  each  has  a  resist- 
ance of  80  ohms.  Another  series  of  4  similar  lamps  is  con- 
nected in  parallel  with  the  first  series.  Find  the  total 
resistance.  Ans.    177.7. 

6.  If  220  volts  are  applied  to  the  above  circuit,  find 
the  current.  Ans.   1.237. 

41.  Computing  Resistance  of  a  Conductor  from  Its  Size 
and  Length.  The  resistance  in  ohms  of  a  conductor  of 
known  material  may  be  computed  from  the  formula, 

Kl 

where,    R  =  resistance  of  the  conductor  in  ohms, 
Z  =  length  of  the  conductor  in  feet, 
A  =  area  of  the  conductor  in  circular  mils, 
K  =  a  constant  depending  on  the  material  in  the 
conductor,  and  is  equal  to  the  resistance  per 
mil  foot  in  ohms. 

One  circular  mil  is  the  area  of  a  circle  1  mil  in  diameter, 
A  mil  is  one  thousandth  of  an  inch. 

A  circle  2  mils  in  diameter  has  an  area  of  4  circular  mils, 
and  a  circle  3  mils  in  diameter  has  an  area  of  9  circular 


CIRCULAR   MILS 


41 


mils.  The  area  of  any  circle  in  circular  mils  is  equal  to  the 
square  of  the  diameter  in  mils.  The  circular  mil  is  a  con- 
venient unit  for  measuring  the  areas  of  wires  since  it  elimi- 
nates the  use  of  tt  (3. 1410). 


1  circular  mil 

4-  circul 

ar  mils 

Fig.  23. 
EXERCISE 

15 

9  circulaj 

-  mils 

Evaluate  the  formula  to  find  the 

missing 

factors: 

R 

A' 

I 

A 

Ans. 

1. 

10.4 

1000' 

16510 

.630. 

2. 

10.4 

275' 

810 

3.525. 

3. 

10.4 

1  mile 

6530 

8.41. 

4.      0.528 

25' 

509 

10.8. 

5.      7.79 

25 

409 

127.4. 

6.    13.27 

10.4 

404 

515'. 

7.      o.o 

10.4 

850' 

1607. 

8. 

10.4 

1  mile 

10381 

5.29. 

9.  Find  the  resistance  of  680  feet  of  No.  25  B.  S.  gauge 
German  silver  wire  (K  =  125,  A  =  320 . 4) .  Ans.  265 . 5. 

10.  Find  the  resistance  of  20  miles  of  trolley  wire  made 
of  No.  00  B.  S.  gauge  copper  wire  (iv  =  10.4,  A  =  133080). 

Ans.  8.26. 


42  EVALUATION 

11.  Find  the  area  of  cross-section  of  copper  wire  having 
a  resistance  of  140  ohms  per  mile.  Ans.  392.2. 

12.  It  is  desired  to  transmit  200  amperes  to  a  point 
2500  feet  from  the  generator  with  not  more  than  4  volts 
line  drop.     What  diameter  copper  wire  must  be  used? 

Ans.    1140  mils. 

13.  It  is  desireu  to  transmit  50  amperes  5  miles  with 
a  line  drop  of  not  more  than  10  volts.  What  size  copper 
wire  must  be  used?  Ans.  Area  =  1372800  circular  mils. 

K1 

14.  Solve  the  formula  -R  =  -r  for  K  and  for  A. 

v    RA     ,     Kl 
Ans.  K—-J-,  A=  —  . 

42.  Resistance   of   a   Conductor    at   Different  Temper- 
atures.     When  the  temperature  of  most  conductors  changes, 
their  resistances  change  also.     The  formula  used  to  find 
resistance  change  with  change  of  temperature  is 
R^Ri+Rtat, 

where,    Rt  =  initial  resistance, 

Rf=  resistance    at    any    other    temperature    higher 
than  the  original  temperature, 
t  =  change  of  temperature  in  degrees  centigrade, 
a  =  temperature  coefficient  of  resistance. 

EXERCISE    16 

Find  the  missing  terms* 


Rf 

Ri 

a 

t 

1. 

200 

.00406 

25 

Ans.  220.3. 

2. 

5000 

.00406 

85 

0725.5. 

3. 

50.8 

10 

.0034 

1200. 

4. 

60 

55 

.00406 

22.39 

5.  220  .00406  25  199.7. 

6.  25  .0034  650  7.78. 


FIELD  INTENSITY  INSIDE  OF  A  COIL  43 

7.  The  resistance  of  a  coil  of  copper  wire  at  39°  is  300 
ohms.  What  will  be  the  resistance  of  the  coil  at  60° 
(a  =.003613).  Ans.  332.76. 

8.  The  resistance  of  a  coil  of  copper  wire  is  200  ohms 
at  40°.     What  will  it  be  at  90°  (a  =  0 .  003605) . 

Ans.  236.05. 

9.  What  will  be  the  resistance  of  a  copper  wire  at  25° 
if  the  resistance  at  48°  is  2 .  08  ohms?     (a  =  0 .  00383 1) . 

Ans.   1.91. 

10.  Solve  the  formula  R/=RiJrRiat  for  a  and  for  t. 

.  Rf—Ri     ,    Rf—Rt 

Ans-a  =  -7^'   ^^aT- 

43.  Field  Intensity  Inside  a  Coil.  The  formula  used  for 
finding  the  field  intensity  inside  a  coil  of  wire  is 

where,  H  —  the  field  intensity  inside  the  coil    measured  in 
gausses, 
N  =  the  number  of  turns  of  wire, 
I  =  current  through  the  coil  in  amperes, 
L  =  length  of  the  coil  in  centimeters. 


EXERCISE 

17 

Find  the  missing  terms: 

H                N              I 

L 

Ans. 

1.                          80              2 

12 

16.8 

2.     40               115               1.5 

5.434 

3.  33.35  1.75  1575  238.2. 

4.  22.31  76  16.34  3.806. 

5.  85  .85  20.1  4.53 

6.  38.5  65.5  .741  1.59 


EVALUATION 


Field  Assembly  of  a  Six-Pole  Direct  Current  Generator  with  Slotted  Poles. 


TOTAL  FLUX  IN  MAGNETIC  CIRCUIT  45 

7.  Find  the  field  intensity  inside  a  coil  of  wire  12  cm. 
long  having  100  turns  and  carrying  a  current  of  0.5  ampere. 

Ans.  5.25. 

8.  A  coil  of  wire  is  to  have  a  field  intensity  of  40  gausses 
and  is  to  be  15  cm.  long,  and  carry  a  current  of  1.2  amperes. 
Find  the  number  of  turns  necessary.  Ans.  396.8. 

9.  A  coil  18  cm.  long  has  75  turns.  What  current  will 
be  necessary  to  give  a  field  intensity  of  (350  gausses? 

Ans.   123.8. 

44.  Total  Flux  in  a  Magnetic  Circuit.  The  following 
formulas  are  given  for  finding  the  total  flux  through  a  mag- 
netic circuit : 

AwNI 


*  =  ■ 


R 


where,   $  =  total  flux  through  the  magnetic  circuit, 

N  —  number  of  turns  of  wire  in  the  magnetizing  coil, 
/  =  current  in  amperes  in  the  magnetizing  coil, 
R  =  reluctance  of  magnetic  circuit,  computed  as  ex- 
plained in  the  following  equation. 
In  the  above  equation,  the  value  of  R,  the  reluctance, 
is  found  as  follows: 

where,  I  =  average  length  of  magnetic  circuit  in  centimeters. 
U.  —  permeability  of  circuit . 
A  =  average  cross-sectional  area  of  magnetic  circuit. 

When  the  circuit  is  made  up  of  different  parts,  as  the 
frame  of  the  motor,  air  gaps,  and  core  of  the  armature, 
Joint  reluctance,  or  R  =  R\-\-Ro-\-Rs~\-  .  .  . 

h      j      h 

fX\A\       /X2-4-2 


46  EVALUATION 

where  R\,  R2  and  Rz  are  reluctances  of  the  different  parts 
of  the  circuit  and  similarly  U,  l>,  etc.,  refer  to  the  lengths 
in  centimeters  of  different  parts  of  the  circuit. 


EXERCISE    18 

1.  Find  the  total  flux  through  a  circuit  composed  of 
iron  and  an  air  gap,  magnetized  by  50  turns  of  wire  carrying 
a  current  of  4  amperes,  if  the  length  of  the  iron  is  36  cm., 
cross-section  area  12  sq.  cm.,  and  /x  =  2000;  and  the  length 
of  the  air  gap  is  2  cm.,  area  15  sq.  cm.,  and  /*=1. 

Ans.   1863. 

2.  Find  the  total  flux  in  a  motor  if  the  path  consists 
of  the  frame  of  the  machine,  the  core  of  the  coils,  two  air 
gaps  between  the  coils  and  armature,  and  the  armature. 
The  magnetizing  current  is  24  amperes  and  has  200  turns. 
The  cores  of  the  machine  have  a  cross  section  of  150  sq. 
cm.,  a  length  (including  both  cores)  of  45  cm.,  and  /x  =  200(). 
The  frame  of  the  machine  has  a  length  of  75  cm.,  an  average 
area  of  175  sq.  cm.,  and  ai=1500.  The  two  air  gaps  have 
an  area  of  180  sq.  cm.,  each  has  a  length  of  1.5  cm.,  and 
p=l.  The  armature  has  an  area  of  160  sq.  cm.,  a  length 
of  40  cm.,  and  M  =  2000.  Ans.  350,079. 

45.  Armature  Winding.  The  following  formula  is  some- 
times used  in  armature  winding. 

y=  -^ — ±2w. 

When  the  positive  sign  is  used  in  the  fraction,  use  the  positive 
sign  before  2m,  etc. 


ARMATURE  WINDING 


47 


Partly  Wound  Armature  showing  Method  of  Assembling  Coils. 


Details  of  Commutator  Construction. 


48  EVALUATION 

EXERCISE   19 

1.  Evaluate  the  formula  of  Section  45,  when  N=  120, 
b  =  18,  p=3,  ra  =  3.  Aris.  29  or  11. 

2.  Evaluate  the  formula  of  Section  45,   when  Ar  =  80, 
6  =  24,  p  =  4,  wz=2.  Ans.  17  or  3. 


CHAPTER  IV 

POSITIVE  AND  NEGATIVE  NUMBERS 

46.  Negative  Numbers.  The  numbers  used  in  arith- 
metic consist  of  a  complete  system  of  numbers  and  fractions, 
ranging  in  value  from  zero  up.  The  order  and  value  of 
these  numbers  ma}'  be  represented  along  a  straight  line, 
as  in  Fig.  24. 


Fig.  24. 

This  system  of  numbers  fails  to  express  completely  some 
numbers  encountered  in  algebra.     For  example: 

1.  In  the  case  of  the  thermometer  there  is  a  5 
above  zero  and  a  5  below  zero,  represented  on  the 
scale  as  in  Fig.  25. 

The  5  above  zero  and  the  5  below  zero  can 
be  distinguished  conveniently  by  +5  (called  posi- 
tive 5)  and  —5  (called  negative  5).  Then  +12 
means  12  above  and  —21  means  21  below  zero.         Fig.  25. 

W  E 


\+S 
0 

H5 


-6  0  +6 

Fig.  26. 

2.  Two  distances  in  opposite  directions,  as  shown  in 
Fig.  26,  can  be  represented  conveniently  by  positive  and 
negative  numbers,  as  +6  and  —6. 

49 


50 


POSITIVE  AND   NEGATIVE   NUMBERS 


Observe  that 

1.  The  (  +  )  (post l ire)  and  (  — )  (negative)  signs  have  a 
new  meaning,  being  used  to  distinguish  things  of  opposite 
nature.  The  (  +  )  and  (  — )  will  continue  to  be  used  as  signs 
of  addition  and  subtraction  as  well  as  signs  of  quality. 

—5     -4     -3    —2    — 1       6     +'l    +2    +  3    +4    +5» 


+3 

+2 

+1 

0 

-1 


Fig.  27. 


2.  77ie  positive  numbers  are  the  same  as  the  numbers  used 
in  arithmetic,  and  when  no  sign  is  expressed  the  (  +  )  sign  is 
understood. 

3.  The  relative  order  and  value  of  positive  and  negative 
numbers  can  be  represented  by  the  scale  of  Fig.  27. 

47.  Addition.  Algebraic  addition  is  the  combination  of 
positive  and  negative  numbers.     For  example: 

1.  A  man  travels  7  miles  east  and  then  3  miles  east 
from  that  point.  This  may  be  represented  as  in 
Fig.  28: 


ALGEBRAIC  ADDITION 


51 


-1      0 


3       4       5       6       7 
Fig.  28. 


9       10 


or 


(+7)  +  (+3)  =  +10. 

2.  A  man  travels  7  miles  west  and  then  3  miles  west 
from  that  point.     This  may  be  represented  as  in  Fig.  29: 

K 1 1 K 

— 3P     — 2     -1       1 

—10  -9     -8    -7     -6    —5    —4    —3     -2     -1       0      +1 

Fig.  29. 
or 

(-7)  +  (-3)=-10. 

3.  A  man  travels  7  miles  east  and  then  3  miles  west  from 
that  point.     This  may  be  represented  as  in  Fig.  30: 


—3       -2    -1 


-1       0 


3        4 
Fig.  30. 


or 


(+7)  +  (-3)  =  +4. 

4.  A  man  travels  3  miles  east  and  then  7  miles  west  from 
that  point.     This  may  be  represented  as  in  Fig.  31: 


-7 


-6    -5    -4    -,3    -2     -1 


or 


-5    -4    -3     -2-10        1 
Fig.  31. 

(+3)  +  (-7)  =  -4. 


52  POSITIVE   AND   NEGATIVE   NUMBERS 

The  problems  above  illustrate: 

1.  (+7)  +  (+3)  =  +10. 

2.  (-7)  +  (-3)  =  -10. 

3.  (+7)  +  (-3)  =  +  4. 

4.  (+3)  +  (-7)  =  -  4. 

where  the  signs  within  the  parentheses  are  signs  of  quality 
and  the  signs  between  the  parentheses  indicate  algebraic 
addition.  From  these  examples  the  following  rule  can  be 
stated : 

Rule.  1 .  To  add  two  numbers  of  like  signs  add  the  numbers 
as  in  arithmetic  and  give  the  residt  the  common  sign. 

2.  To   add   two   numbers   of  opposite   signs   subtract   the 

smaller  from  the  larger  and  give  to  the  result  the  sign  of  the 

larger. 

EXERCISE   1 

Add 

1.  +5,   +12.  Ans.   +17. 

2.  -5,   -12.  -17. 

3.  +5,   -12.  -7. 

4.  -5,   +12.  +7. 

5.  -29x,   Ux.  -I5x. 

6.  -15d,  S2d.  lid. 

7.  —3.712/,   -5.23y.  -8.94?/. 

8.  +9.21,   -4.356.  4.854. 

9.  +22,   -13,   +24,   -8.  +25. 

Note  to  Prob.  9.  This  problem  can  be  solved  most  efficiently 
by  adding  all  the  positives  and  all  the  negatives  and  combining  results. 

10.  +34,   -45,   -17,   +12.  Ans.   -16. 

11.  5x,   -7x,  Ux.  \2x. 

12.  -13.T2,   -5x2,  29.T2.  Ux2. 

13.  (a3+3a2b+3a&2+63)  +  (2a3+4a26-7«62) 

+  (-5a3-a62+463). 


SUBTRACTION  53 

Solution.     Arrange  with  similar  terms  in  a  column  and 
add  columns  thus: 

cP+3a2b+ZdP+  63 
2a3+4a26-7a&2 
-5a3  -  ao2+463 


-2a3+7a26-5a62  +  563 


14.  (5x2  -  3xy  +  2y2)  4-  (2x2 + 3xy  -  y2)  +  (x2  -  4xy) . 

Ans.  8x2  —  Axy-\-y2. 

15.  (3a  4- 56  -  4c)  4- (2a  +  3c)  +  (a -126).     Ans.  6a- 76- c. 

16.  (oax2  -  3aij2+2bz2)  +  (7a?/  -  6a.x2) 

4-(8bz2+2ax2-ay2).  Ans.  az2+3a?/2-f-106z2. 

48.  Subtraction.  Subtraction  means  to  find  the  differ- 
ence between  two  numbers,  that  is,  the  distance  between 
the  two  numbers  on  the  number  scale.  By  reference  to  the 
number  scale  for  positive  and  negative  numbers  find: 

1.  The  distance  to  +5  from  +3.     Fig.  32. 


-101234567 
Fig.  32. 

This  is  seen  to  be  2  in  the  positive  direction.  It  may 
be  computed  by  subtracting  the  second  number  from  the 
first.     Therefore 

(4-5) -(+3)  =  4-2. 

2.  The  distance  to  +3  from  +5.     Fig.  33. 


-10+1+2     3      4      5 
Fig.  33. 


54  POSITIVE  AND  NEGATIVE  NUMBERS 

This  is  seen  to  be  2  in  the  negative  direction.  It  is 
equivalent  to  subtracting  the  second  number  from  the  first. 
Therefore 

(  +  3)-(  +  5)=-2. 

3.  The  distance  to  -3  from  +5.     Fig.  34. 


-3-2-10       123       45       6 

Fig.  34. 

This  is  seen  to  be  8  in  the  negative  direction.  It  is 
equivalent  to  subtracting  the  second  number  from  the  first. 
Therefore 

(_3)-(+5)  =  -8. 

4.  The  distance  to  +5  from  -3.     Fig.  35. 


-3     -2     -1       0        1        5        5 I       5        6 
Fig.  35. 

This  is  seen  to  be  8  in  the  positive  direction.  It  is 
equivalent  to  subtracting  the  second  number  from  the  first. 
Therefore 

(+5)  -(-3)  =  +8. 


EXERCISE   2 

1.  Find  the  following  distances: 

To        -5 

+  5 

-10 

+3 

-8 

+7 

From   +5 

-5 

-  3 

-8 

+3 

+4 

2.  Add 

-5 

+5 

-10 

+3 

-8 

+7 

-5 

+  5 

+  3 

+8 

-3 

-4 

SUBTRACTION  55 

Compare  the  results  in  the  two  problems  above  and  note 
the  following  rule : 

Rule.  To  subtract  one  number  from  another  change  the 
sign  of  the  subtrahend  and  apply  the  rules  for  addition. 

EXERCISE  3 


Ans.  12. 


Subti 
1. 

■act: 

+32 
+  20 

2. 

-24 

-18 

3. 

-42 
+  16 

4. 

+  18 
-22 

5. 

+  17:r2 
+  bx2 

6. 

-29ax 
+  16ax 

7. 

-\-hay2 
-lay2 

8. 

+2a-36 
+  a+  b 

-58. 
+40. 

12x2. 

—45ax. 

Ylay2. 


a-  46. 

9.  (-5)-(-7).  2. 

10.  (+8)-(-3).  11. 

11.  (-12)-(+16).  -28. 

12.  (3x+7y)-(2x+Sy).  x+4y. 


56 


POSITIVE   AND    NEGATIVE   NUMBERS 


13.  (Zri-4xy+7y')  -  (2a?+3xy-  I2y2). 

Ans.  x2  —  7xy-\-l9y2. 

14.  (5a2  +  6a6  +  2b2)  -  (3a2  -  lab  +  b2)  -  (2a2  -  Sab + 362) . 

Ans.  (k/2  +  12ao. 

15.  (\2x3-7x2+4x) - (2x3+3x2-8x)  +  (4.T3-rvr-,  +  6a;). 

Ans.   14a:3-15.r2  +  18a;. 
1G.  (a+b)-(c-d).  Ans.  a+6-c+d. 

Note.     Subtraction   will   remove   the   parentheses  but   there    a  re 
no   similar  terms  that  can  be  combined. 


17.  x2+3x-(-5+y). 

Ans.  x2  +  3x4-5  — #. 

18.  4+3x-(2s-5). 

9+x. 

Solve  and  check: 

19.  10.r-(3:c-4)  =  (4x+4)+7. 

2i 

Note.     First  remove  the  parentheses  by  changing  the  signs  of  the 
subtrahend  thus: 

10x-3x+4  =  4.r+4+7. 


20.  6-(3w-4)  =  (2n-3)-(n-l). 

21.  (2x+3)-(-3x-2)  =  25. 

22.  (5x-3)-7=(2x+5). 


Ans.  3. 
4. 
5. 


7T 


Ft 


Fig.  36. 


49.  Multiplication.  When  a  force  is  applied  to  a  lever, 
it  will  cause  the  lever  to  turn  or  to  have  a  tendency  to  turn. 
If  the  force  is  doubled,  the  tendency  to  turn  will  be  doubled; 
or  if  the  length  of  the  lever  is  doubled,  the  tendency  to 
turn  will  be  doubled.  The  tendency  of  a  lever  to  turn  is 
called  the  turning  moment  or  leverage.  The  length  of  the 
lever  from  the  fulcrum  (the  turning  point)  to  the  force  is 
called  the  lever  arm.     See  Fig.  36. 


LAW  OF  SIGNS  57 

The  leverage  caused  by  a  force  is  equal  to  the  force  times 
the  arm. 

An  arm  to  the  right  of  the  fulcrum  is  considered  positive 
and  an  arm  to  the  left  of  the  fulcrum  is  considered  negative. 
An  upward  pulling  force  is  considered  positive  and  a  down- 
ward pulling  force  is  considered  negative.  These  assump- 
tions are  the  same  as  the  numbers  on  the  number  scale. 


Fig.  37. — Positive  Leverage. 

A  positive  force  on  a  positive  arm  must  give  a  positive 
leverage  since  (+4)(4-7)  =  +21  (Arithmetic).     See  Fig.  37. 

Note.  A  leverage  in  the  direction  opposite  to  the  motion  of  the 
hands  of  a  clock  (counter  clockwise  direction)  is  a  positive  leverage. 
Then  a  leverage  in  the  clockwise  direction  must  be  considered  negative. 

A  negative  force  on  a  negative  arm  will  cause  the  lever 
to  turn  in  the  same  direction  (counter  clockwise)  giving 
likewise  a  positive  leverage,   Fig.   38.     Therefore 

(-3)(-7)  =  +21. 


71 

Fig.  38.  — Positive  Leverage. 
A  negative  force  on  a  positive  arm  will  cause  the  lever 


58  POSITIVE  AND   NEGATIVE   NUMBERS 

to  turn  in  the  opposite  direction  (clockwise)  giving  a  nega- 
tive leverage,   Fig.  39.     Therefore 

(+3)(-7)--21. 


7X 

If 

Fig.  39. — Negative  Leverage. 

A  positive  force  on  a  negative  arm  will  cause  the  lever 
to  turn  in  the  clockwise  direction,  Fig.  40.     Therefore 

(-3)(+7)=-21. 


Fig.  40. — Negative  Leverage. 

The  four  preceding  examples  show  that 

1.  (+7)(+3)  =  4-21. 

2.  (-7)(-3)  =  +21. 

3.  (-7)(+3)  =  -21. 

4.  (+7)(-3)=-21. 

From  these  problems  a  law  of  signs  for  multiplication 
can  be  stated. 

50.  Law  of  Signs  for  Multiplication. 

1.  If  two  factors  have  like  signs  their  product  is  positive. 

2.  If  two  factors  have  unlike  signs  their  product  is  negative. 


LAW  OF  EXPONENTS  59 


EXERCISE  4 

Multiply: 

1.  (3)(  +  12). 

Ans.   +36. 

2.  (-3X-12). 

+36. 

3.  (-!)(+!). 

i 

2* 

4.  (2.5)(-4). 

-10. 

5.  (-5.6)(-2, 

3). 

12.88. 

6.  (3f)(-5). 

m 

51.  Law  of  Exponents.     By  definition  of  an  exponent 

x3  means  xxx 
and 

r1  means  xxxx. 
Therefore 

(x3)  (x4)  =  (xxx)  (xxxx)  =  X7. 

From  this  and  similar  problems  a  law  of  exponents  can 
be  stated: 

Rule.  To  multiply  powers  of  the  same  base  add  their 
exponents. 

EXERCISE  5 

Multiply: 

1.  x7x4.  Ans.  xn. 

3.   (a3)(-o2).  -a5. 

3.  (-v2)(-y).  +y3. 

4.  x3x5x.  x9. 

52.  Multiplication  of  Monomials.  When  different  bases 
occur  in  the  factors,  their  product  can  be  indicated  only. 
Thus 

(x3)(if)=x3y2. 


60  POSITIVE  AND   NEGATIVE   NUMBERS 

When  several  different  bases  occur  in  each  factor,  as 

(3a2&3r)(4a3t4a;6), 

only  the  powers  of  the  same  base  can  be  combined.     For 
example, 

(3a263r)  (4a36%6)  =  3  X4a2a36364r:r6  =  I2a5b7rx*. 

Note.  Expressions  whose  parts  are  not  separated  by  the  (+)  or 
(  — )  signs  are  called  monomials. 

From  the  above  example  a  rule  for  the  multiplication  of 
monomials  can  be  stated. 

Rule.  To  multiply  two  monomials  multiply  their  numerical 
coefficients  and  annex  all  the  different  bases,  giving  to  each  base 
an  exponent  equal  to  the  sums  of  the  exponents  of  that  base  in 
the  two  factors. 


EXERCISE  C 

Multiply: 

1.  (4x3)(5a:4). 

Ans.  20x7. 

2.  (I2x2y3)(2x3y5) 

24x5/y8. 

3.  (5a3xV)(6a2a;22). 

30a5zV22 

4.  (-2ab2)(lZb2c). 

-  26a64c. 

5.  (5xy)(  —  4xz). 

-  20x2yz. 

53.  Law  of  Leverages.     If  two  forces  act  upon  the  same 
lever  at  the  same  time,  the  lever  will  be  in  balance  when  the 


BALANCED  LEVERS 


61 


positive  leverages  equal  the  negative  leverages;  that  is, 
when  the  sum  of  all  the  leverages  equals  zero.  Thus  in 
Fig.  41: 


+  8 


Fig.  41 


12 


The  leverage  caused  by  the  force    8  =  (  —  3)  (  —  8)  =  +24. 
The  leverage  caused  by  the  force  12=(— 12)(-(-2)=  —24. 

(+24)  +  (-24)=0, 
0  =  0. 

Therefore  the  lever  will  balance. 


EXERCISE   7 

1.  Find  the  sum  of  all  the  leverages  in  Fig.  42. 


« 

h 

L 

\     } 

•3 

6*' 

Fig.  42. 


Will  the  lever  balance?     Why? 

2.  Find  the  value  of  x  that  will  make  the  levers  balance 
in  Figs.  43  (a)-43  (e). 


62 


POSITIVE  AND  NEGATIVE  NUMBERS 


A" 


17 


t«-i-*i 


(a) 


-5.5- 


-3.3- 


125 


<M 


-2 >k- 


^1T"^ 


(c) 


(d) 


{20 

(•) 

Fig.  43. 


>U — 9 


rlO 


Solution. 

4z  4-2-50  =  0 
4z  =  48 
x=12. 


Ans.  75. 


T^^ 


Ans.  10. 


Ans.  6f. 


Ans.  35. 


54.  Multiplication  of  Polynomials  by  Monomials.     The 

perimeter  of  a  rectangle  is  2 (a -\-b) 
and  also  2a  4- 26.    Fig.  44. 
Therefore 

2(a+b)  =  2a+2b. 


b 

Fig.  44. 


Expressions  like  (a  4- 6)  and 
(x2  —  3x 4-4)  which  consist  of  two 
or  more  parts  added  or  subtracted,  are  called  polynomials. 


MULTIPLICATION   OF   POLYNOMIALS 


63 


From  the  above  problem  a  rule  for  multiplication  of 
a  polynomial  by  a  monomial  can  be  stated: 

Rule.  To  multiply  a  polynomial  by  a  monomial  multiply 
each  term  of  the  polynomial  by  the  monomial. 


EXERCISE  8 

Multiply: 

1.  a2+3a&+462  by  5. 

2.  x2  —  3xy+4y2  by  5x. 

3.  a3-a2b-2ab2  by  -2ab. 


4.  x2  —  7x2y2+4y2  by  3x2y3. 


Ans.  5a2+15a6+2062. 
5r3  —  1  bx2y  -\-2Qxy2. 
-2a46+2a362+4a263. 
Zx4y3-21x4y5+12x2y5. 


xy 

6x 

27/ 

12 

55.  Multiplication   of   a   Polynomial   by   a   Polynomial. 

The  product  (y+6)(a;  +  2)  can 

be    represented    by  the   rect-  V 

angle,    Fig.  45,   whose   length 

is    y-\-Q    and    whose     width 

is    x-\-2.      The    area    of    the 

rectangle      is      the      product 

(s/+6)(x+2).      The     area    is 

also  the  sum  of  the  areas  of  Fig.  45. 

the    four     small     rectangles, 

that  is,   xy-{-Qx-\-2y+l2.     Therefore 

(2/+6)(z+2)=:cy4-6z+2?/+12 

From  the  above  problem  the  following  rule  can  be 
stated : 

Rule.  To  multiply  a  polynomial  by  a  polynomial  mul- 
tiply each  term  of  the  one  by  every  term  of  the  other  and  combine 
the  results. 


64  POSITIVE  AND   NEGATIVE   NUMBERS 

The  work  can  be  arranged  conveniently  thus: 


x  -2 

x  +  3 

x2  —  2x        (multiplying  x— 2  by  x\ 

3x— 6  (multiplying  x  —  2  by  3) 

ar  +  x  —  6  (combining  similar  terms) 

EXERCISE   9 

Multiply: 

1.  (x+3)(s+4). 

Ans.  z2+7z+12. 

2.  (*-5)(s-2). 

x2-7z+10. 

3.  (x-y)(x-y). 

x2  —  2.T4/+?/2. 

4.  (2a +36)  (3a- 

46).                              6a2  +  a6-12&2 

5.   (a -6)  (a -6). 

a2-2a6  +  62. 

6.  (z  +  2)(x  +  2). 

z2+4z+4. 

7.   (y-S(y-S). 

?/2  — 6?/+9. 

8.  (n-5)(n-5). 

w2-10n+25. 

56.  Division.  The  division  of  positive  and  negative 
numbers  requires  a  law  of  signs  and  a  law  of  exponents 
which  are  developed  by  a  study  of  the  laws  of  signs  and 
exponents  for  multiplication.  Division  is  the  opposite  of 
multiplication,  hence, 


Since  (  +  5)(+6)  =  +30  it  follows  (+30) 
Since  (+5)  (-6)=  -30  it  follows  (-30) 

Also  (-30) 
Since  ( -  5)  ( -  6)  =  (+30)  it  follows  (+30) 


(+6)  =  +5. 
(-6)  =  +5. 
(  +  5)= -6. 
(-0)  =  -5. 


From  these  problems  a  law  of  signs  for  division  can  be 
stated : 


DIVISION  65 

57.  Law  of  Signs  for  Division. 

1.  //  two   numbers   have    like    signs    their    quotient    is 
positive. 

2.  //  two   numbers   have   unlike   signs   their   quotient   is 
negative. 

EXERCISE    10 

Divide : 

1.  (  +  25)  4- (+5).  Ans.   +5. 

2.  (-63)  -K-3).  +21. 

3.  (-24)h-(+5).  -4.8. 

4.  (+38)-*-(-4).  -9.5. 

(-4).  +9. 


5.  (-36) 

6.  (-24) 


(-3). 


58.  Exponents  in  Division.     By  the  law  of  exponents 
for  multiplication, 

x7x*  =  x11 
Therefore 

X11  +x7  =  x4 
and 

XU  -trX4:=X7 

From  this  problem  the  following  law  of  exponents  for 
division  can  be  stated: 

1 .  To  divide  powers  of  the  same  base  subtract  the  exponent 
of  the  divisor  from  the  exponent  of  the  dividend. 

2.  The  quotient  of  powers  of  different  bases  can  be  indicated 
only. 

59.  Division  of  Monomials. 

Example.     Divide  24a3rVz4  by  Sa2bx2y5xz6. 


66  POSITIVE   AND   NEGATIVE   NUMBERS 

Solution. 

3    a         y* 
M<P  pPft*     _Sayz 
%  fi2  b  #?  y&  t*~  bz2 

Rule.     To  divide  a  monomial  by  a  monomial  write  the  two 
monomials  as  a  fraction  and  cancel  equal  factors. 


EXERCISE 

11 

Divide : 

1.  Ylo?x  by  3a. 

Ans. 

4a2x. 

2.  'S5a2x7y8  by  -7a2x3y6. 

—  5.T4?/2. 

3.   —  46a365m3  by  4a3bm. 

— 11 .  .")/>'/// 

4.   -75x7b4c3  by  -10x&4. 

7 .  5x6c?. 

5.  38r2s3c  by  -  19/-Vc. 

—  2. 

6.  27x3  by  9 if. 

3-". 

60.  Division  of  a  Polynomial  by  a  Monomial.     By  the 
rules  for  multiplication 

5  (x  -2y)  =  5x-10y 
Therefore : 

(5x-10y)+5=x-2y 

Rule.     To  divide  a  polynomial  by  a  monomial  divide  each 
term  of  the  polynomial  by  the  monomial. 

The  work  can  be  arranged  conveniently  thus: 

lOsV-  15sY  +  25sV  =  2     3  _  3^+5^ 
5xy/- 


DIVISION  67 


EXERCISE    12 

Divide : 

L  r3-2z2+a  ^    aa_2x+lm 

x 

2.  °y2-^+4q.  </2-3y+4. 

3.  *=**.  l-y. 

x 

.    Ri-\-Ridt 

4.  5 .  l+at. 

K    7a3x2-14a2x34-28a5  „  ,  rt  _      .  , 

5. — =-= —     — .  —  ax-+2xi  —  4a6. 

—  7cr 

6.  ^^-l^3-!^^  2*^-4^-5^. 


CHAPTER  V 
RATIO  AND  PROPORTION 

61.  Ratio.  The  relation  of  one  number  to  another 
number  of  the  same  kind  is  called  a  ratio.  A  ratio  is 
expressed  as  5:3,  read  the  ratio  of  5  to  3,  or  as  f.  The 
value  of  a  ratio  is  found  by  writing  it  as  a  fraction  and 
reducing  the  fraction  to  its  lowest  terms. 

EXERCISE    1 

1.  Find  the  value  of  the  ratio  f|. 

Solution.     ff=f 

7X9 

2.  Find  the  value  of  the  ratio  ^tt- 

Solution.     »  — -=  =  g  =  1  s 

3.  Find  the  ratio  of  the  areas  of  two  rectangles  having 
dimensions  15  by  18  and  24  by  12. 

3 
5       9 
#X#         15 


Solution. 


24  xn         16 


68 


PERCENTAGE  69 

4.  Find  the  ratio  of  the  circumferences  of  two  circles 
having  diameters  of  3"  and  8".  Ans.  |. 

5.  Find  the  ratio  of  the  areas  of  two  circles  having  radii 
of  7"  and  12".  Ans.  ffi. 

6.  Find  the  ratio  of  the  two  values  of  W  in  the  formula 

W=.2Qbh, 

when  b  =  18,  h  =  25,  and  when  6=15,  h  =  42.  Ans.  f. 

62.  Percentage.  It  is  often  more  significant  to  know  the 
relative  number  than  to  know  the  absolute  number.  For 
example,  a  city  may  increase  in  population  25,000  in  ten 
years,  but  it  is  more  significant  of  the  growth  of  a  city  to  say 
that  it  increased  in  population  60  per  cent  or  10  per  cent. 
When  a  chemist  makes  an  analysis  he  records  his  results 
as  18  per  cent  iron  in  the  ore  he  analyzes,  rather  than 
25  grams  of  iron  in  the  sample  he  analyzed.  The  relative 
number  is  called  per  cent  and  is  found  by  taking  the  ratio 
of  a  part  to  the  whole.  For  example,  if  25  grams  of  iron 
are  found  in  175  grams  of  ore,  then  the  ratio  which  gives 
the  per  cent  of  iron  in  the  ore  is 

grams  of  iron      „  _       .       .  .  _  .       .  ,  , 

— = =i%  =  y  =  .  142+  =  14+  percent, 

grams  ol  ore 

Observe  that: 

1-  TT5,  r,  .142+,  and  14  per  cent  are  different  ways  of 
expressing  the  same  ratio. 

2.  Per  cent  is  expressed  on  the  basis  of  hundredths,  the 
ratio  .14  means  14  per  cent,  the  ratio  .855  means  85|  per 
cent. 


70  RATIO   AND   PROPORTION 


EXERCISE   2 


1.  If  62^  tons  of  iron  are  obtained  from  625  tons  of  ore, 
what  is  the  per  cent  of  iron  in  the  ore?  Ans.   10. 

2.  If  3  gallons  of  water  are  added  to  22  gallons  of  alcohol, 
what  is  the  per  cent  of  alcohol  in  the  mixture?         Ans.   12. 

3.  Potassium  chloride  is  composed  of  39  parts  of  potas- 
sium to  35.5  parts  chlorine.  Find  the  percentage  of  chlo- 
rine in  potassium  chloride.  Ans.  47. 

4.  Babbitt  metal  is  composed  of  92  parts  of  tin  to  8  parts 
of  copper  and  4  parts  of  antimony.  Find  the  percentage 
of  copper  in  babbitt  metal.  Ans.  8. 

5.  If  5  lbs.  of  vegetables  lose  5  oz.  in  drying,  what  part 
of  the  original  weight  was  water?  Ans.  6  per  cent. 

6.  An  ore  analyzed  23  per  cent  iron.  How  many  lbs.  of 
iron  could  be  obtained  from  1  ton  of  ore? 

Solution.     If  x  lbs.  could  be  obtained, 

Then  ^  .23 

a;  =  460 

7.  If  water  is  87|  per  cent  oxygen,  how  many  grams  of 
oxygen  could  be  obtained  from  17  grams  of  water? 

Ans.  14f. 

8.  If  potassium  nitrate  is  39  per  cent  potassium,  how 
many  grams  of  potassium  nitrate  would  be  required  to  give 
24  grams  of  potassium?  Ans.  61 . 5. 

9.  If  ore  contains  14  per  cent  iron,  how  many  tons  of 
ore  are  required  to  get  125  fons  of  iron?  Ans.  892.86. 

10.  If  2  tons  of  coal  give  625  lbs.  of  ash,  wha  per  cent 
of  the  conl  is  ash?  A,  s.  16. 


EFFICIENCY  71 

63.  Efficiency.  The  efficiency  of  a  machine  is  the  ratio 
of  the  work  done  by  the  machine  to  the  work  put  into  the 
machine.  Efficiency  is  expressed  as  per  cent.  The  rule  in 
the  formula  form  is, 

-Fa;  •         _  output  _  input  — losses  _        output 


input  input  output + losses 

EXERCISE  3 

Find  the  missing  values  in  the  following: 


LOSSES 

INPUT 

EFFICIENCY 

1. 

950 

6850 

Ans.  86% 

2. 

6950 

75 

1737. 

3. 

692 

6820 

89%. 

4. 

800 

90 

8000. 

5.  The  losses  of  a  motor  while  running  are  500  watts, 
the  input  is  4500  watts,  what  is  the  efficiency  of  the  machine? 

Ans.  89  per  cent. 

6.  The  efficiency  of  a  motor  is  79  per  cent.  The  input  is 
60,000  watts,  what  is  the  output?  Ans.  47,400. 

7.  A  certain  acid  mixture  requires  2  parts  of  acid  to 
3  parts  of  water.  Find  what  per  cent  of  the  mixture  is  acid 
and  what  per  cent  is  water.  Ans.  40,  60. 

8.  In  the  above  problem  find  how  much  acid  and  how 
much  water  are  in  14  gallons  of  the  mixture.    Ans.  5.6,  8.4. 

64.  Separating  in  a  Given  Ratio.  Problem  8  above 
could  be  solved  more  readily  as  follows: 

Let  2x  =  amount  of  acid 


72  RATIO  AND   PROPORTION 

Then  2>x  =  amount  of  water 

2x+3z=14  gallons 
5x=U 
x=   2| 
2x  =   5f  gallons  of  acid 
2>x  =  8f  gallons  of  water 

EXERCISE  4 

1.  Divide  35  into  two  parts  in  the  ratio  of  2  to  3. 

Ans.  14,  21. 

2.  Divide  180  in  the  ratio  4  to  5.  Ans.  80,  100. 

3.  Bronze  is  composed  of  11  parts  of  tin  to  39  parts  of 
copper.  Find  the  number  of  lbs.  of  tin  and  copper  in 
625  lbs.  of  bronze.  Ans.  137 . 5,  487 . 5. 

4.  Eighteen  carat  gold  is  composed  of  18  parts  of  pure 
gold  to  6  parts  of  other  metal.  How  much  pure  gold  is 
contained  in  4.8  oz.  of  the  alloy?  Ans.  3.6. 

5.  In  an  electric  circuit  the  fall  of  potential  over  any 
two  of  the  parts  is  in  the  ratio  of  the  resistances.  24  ohms 
and  40  ohms  are  joined  in  a  circuit.  The  drop  of  the  poten- 
tial over  both  resistances  is  90  ohms,  find  the  drop  over 
each  resistance.  Ans.  33f ,  56|. 

6.  In  a  divided  circuit  the  current  divides  in  the  ratio 
of  the  resistances,  the  larger  current  taking  the  path  of  the 
smaller  resistance.  If  the  two  resistances  of  a  divided 
circuit  are  24  ohms  and  36  ohms,,  and  the  total  current 
flowing  is  8  amperes,  find  how  much  current  will  flow  in 
each  branch.  Ans.  4.8,  3.2. 

65.  Proportion.  In  two  samples  of  the  same  kind  of 
bronze  the  ratios  of  the  copper  to  the  tin  are  equal.  When 
two  ratios  are  equal  chey  form  a  -proportion.     A  proportion 


PROPORTION 


73 


Wheatstone  Bridge. 


Leeds  and  Northrup  Potentiometer. 


74 


RATIO  AND  PROPORTION 


is  written  as  f  =  -rr,  and  read  the  ratio  of  3  to  7  equals  the 
ratio  of  9  to  21;  or  3  :  7  ::  9  :  21,  and  read  3  is  to  7  as 
9  is  to  21.  The  first  and  last  terms  of  a  proportion  are 
called  extremes  and  the  second  and  third  are  called  means. 

A  proportion  is  used  in  finding  ratios  of  quantities  when 
some  other  ratio  between  the  same  two  quantities  is  known. 

For  example:  In  bronze  11  parts  of  tin  combine  with 
39  parts  of  copper.  The  same  ratio  holds  for  any  quantity 
of  bronze.  How  many  parts  of  tin  will  combine  with 
260  lbs.  of  copper? 

Solution.     Let  rr  =  the  number  of  pounds  of  tin. 

11  -.      x 

Then  the  ratio  ttk  =  the  ratio  ^^ 
39  200 

11=  x 

39  ~~  260 
2860  =  39s 

z=73.33+  lbs.  of  tin 
Observe  that: 

1.  A  proportion  is  an  equation. 

2.  A  proportion  can  be  cleared  of  fractions  by  multiplying 
both  members  by  the  product  of  the  denominators. 

3.  To  clear  a  proportion  of  fractions  write  the  product  of 
the  means  equals  the  product  of  the  extremes. 


66.  The  Wheatstone  Bridge 
the  unknown  resist- 
ance is  placed  in  a 
Wheatstone  bridge, 
as  shown  in  Fig.  46, 
where  any  three  of 
the  resistances  are 
k  n  o  w  n  a  n  d  .the 
fourth  resistance  is 
the  resistance  to  be 


In  measuring  resistance 


Fig.  46. — Wheatstone  Bridge. 


FRACTIONS  OF  A  GIVEN  DENOMINATOR  75 

measured.     The   three   known   resistances   are   adjusted   to 

/?       /?• 
make  yt  =  ~d^-     The    galvanometer    G    indicates   when    the 
/12     Ra 

resistances  are  so  adjusted  that  the  ratios  are  equal.     The 

proportion  ~-  =  ■=-  is  used  as  a  formula. 
Ro      R± 

EXERCISE   5 

Find  the  missing  numbers  in  the  following: 

Ri  R2  R3  R*  Ans. 

1.  5  4  3  6.67 

2.  1.5                          3.5  2.5  1.07 

3.  9.85  6.8          9.9  6.84 

4.  1.5  2.35  5  3.19 

5.  8.7                          9.5  45  41.5 

6.  160  32            43  8.6 

7.  650  80            160  19.7 

8.  60                            235  25  6.38 

9.  4.75                        6.843  5  3.47 

10.  275  6.625         9.375  195.1 

11.  If  ^  =  100  and  #4  =  4.26,  find  tf3.  Ans.  426. 

Ri 

67.  Reduction  to   Fractions   of   a   Given  Denominator. 

It   is  often   necessary  to  express  a  fraction   or  a   decimal 
in  halves,  fourths,  sixteenths,  etc. 

Example.     How  many  eighths  in  f  of  an  inch? 

Solution.     If  there  are  x  eighths,  then 

x  _A 

8~5 

5z  =  32 
z  =  6.4 


76  RATIO    AND    PROPORTION 


EXERCISE   6 

1.  How  many  04ths  in  f  of  an  inch?         Ans.  38.4. 

2.  How  many  32ds  in  two  °f  an  inch?  23. 

3.  How  many  64ths  in  .365  of  an  inch?  23.36. 

4.  How  many  16ths  in  .82  of  an  inch?  13.12. 

5.  How  many  64ths  in  TV  of  an  inch?  37.3. 

6.  How  many  thousandths  in  ^T  of  an  inch?  15.625. 

7.  How  many  thousandths  in  t\  of  an  inch?  187.5. 

68.  Writing  a  Proportion.  In  writing  a  proportion  two 
types  of  problems  will  be  encountered.  First,  when  the 
two  quantities  are  so  related  that  an  increase  or  a  decrease 
in  one  will  produce  the  same  kind  of  a  change  on  the 
other. 

For  example:  If  20  men  assemble  8  machines  in  a  day, 
more  men  could  assemble  more  machines,  and  less  men 
less  machines.     This  is  called  a  direct  proportion. 

Secondly,  when  two  quantities  are  so  related  that  an 
increase  or  a  decrease  in  the  one  will  produce  the  opposite 
change  in  the  other. 

For  example:  It  takes  30  men  12  days  to  assemble  a 
machine,  more  men  could  do  it  in  less  time  and  for  less 
men  it  would  take  more  time.  This  is  called  an  inverse 
proportion. 

Example  1.  12  feet  of  angle  iron  weighs  44  lbs.,  how 
much  will  30  ft.  weigh? 

0  ,     .  12  ft     44  lbs. 

solution. 


30  ft      x  lbs. 
Solve  the  equation  for  x.  Ans.   110. 

Every  ratio  must  be  «  comparison  of  similar  things,  and 


DIRECT   AND    INVERSE    PROPORTION  77 

in  every  proportion  both  ratios  must  be  written  in  the  same 
order  of  value,  that  is: 

Small  length     Small  weight 

Large  length     Large  weight 

or 

Large  length  _  Large  weight 

Small  length     Small  weight 

Example  2.     If  30  men  do  a  piece  of  work  in  12  days, 
how  long  will  it  take  20  men  to  do  the  same  work? 
Solution. 

Large  number  of  men     Large  number  of  days 
Small  number  of  men  ~  Small  number  of  days 

That  is 

30=^ 

20~12 

Solve  for  a.  Ans.  18. 

Observe  that: 

1.  In  the  direct  proportion  the  44  #>s.  corresponds  to  12  ft. 
and  x  lbs.  to  30  ft.  and  the  corresponding  numbers  are 
arranged  directly  across  from  each  other  thus: 

12  <     _ S"44 

30  < >x 

2.  In  the  inverse  proportion  the  30  men  correspond  to  12 
days  and  20  men  to  x  days  and  the  corresponding  numbers 
are  arranged  diagonally  thus: 


>fc^><>^r2 


78  RATIO   AND   PROPORTION 


EXERCISE  7 


Solve  by  proportion: 

1.  If  a  steel  rail  3  ft.  long  weighs  112  lbs.,  how  much 
will  a  rail  20  ft.  long  weigh?  Ans.  746.7. 

2.  20  men  do  a  piece  of  work  in  54  days.  How  many 
men  would  it  take  to  do  it  in  30  days?  Ans.  36. 

3.  The  volume  of  a  quantity  of  gas  is  inversely  pro- 
portional to  the  pressure  upon  it.  If  a  quantity  of  gas 
measures  350  cu.  ft.  at  15  lbs.  pressure,  how  many  cubic 
feet  will  it  measure  at  25  lbs.  pressure?  Ans.  210. 

4.  The  pressure  of  627  cu.  ft.  of  gas  is  to  be  reduced 
from  35  lbs.  to  24  lbs.,  what  will  be  the  volume. 

Ans.  1164.37. 

5.  The  volume  of  a  gas  is  directly  proportional  to  the 
absolute  temperature  when  the  pressure  is  constant.  If  a 
quantity  of  gas  occupies  125  cu.  ft.  at  278°,  what  will  be 
its  volume  at  316°?  Ans.  124.09. 

6.  If  a  quantity  of  gas  occupies  28  cu.  ft.  at  a  tempera- 
ture of  250°,  what  will  be  the  temperature  when  it  occupies 
30  cu.  ft.?  Ans.  267.85,. 

7.  If  the  resistance  of  25  ft.  of  wire  is  11.2  ohms,  what 
will  be  the  resistance  of  83  ft.  of  the  same  wire? 

Ans.  37.1841 

8.  An  investment  produces  an  income  of  $600  at  3? 
per  cent.     What  would  it  produce  at  5  per  cent? 

Ans.  857.14. 


CHAPTER  VI 

CUTTING  SPEED,  PULLEYS  AND  GEARS 

69.  Rim  Speed.  When  work  is  turned  in  a  lathe,  the 
work  must  pass  by  the  point  of  the  cutting  tool  at  a  speed 
which  will  complete  the  work  in  the  shortest  possible  time 
without  injury  to  the  work  or  the  tools  used.  When  the 
work  is  turned  one  complete  revolution,   a  point  on  the 

surface  of  the  work 
travels  a  distance  equal 
to  the  circumference  of 
the  work.  In  one  min- 
ute the  point  would 
travel  a  distance  equal 
to  the  circumference 
Fig.  47.  of   the  work    times   the 

number  of  revolutions 
per  minute  (R.P.M.).  The  same  applies  to  emery  wheels, 
grindstones,  and  pulleys.  The  distance  traveled  in  one 
minute  by  a  point  on  the  circumference  of  any  revolving 
object  is  called  rim  speed,  surface  speed  or  cutting  speed. 
Rim  speed  must  be  expressed  in  feet  per  minute. 

Rule.     To  find  rim  speed  multiply  the  circumference  of 
the  revolving  object  by  the  number  of  revolutions  per  minute. 
By  formula: 

RS=C-  (R.P.M.) 

79 


SO  CUTTING  SPEED,    PULLEYS   AND   GEARS 

where  C  =  circumference  in  feet,  and  RS  =  rim  speed.     Or 

C- (R.P.M.) 


KS  = 


12 


where  C  =  circumference  in  inches. 

But  since  C  =  ird,  where  d  is  the  diameter, 

7Tf/(R.P.M.) 


RS= 


12 


Example  1.  The  diameter  of  a  bolt  being  turned  in  a 
lathe  is  3  ins.  If  it  makes  160  R.P.M.,  what  is  the  rim 
speed? 

Solution. 

na      7Td(R.P.M.) 

Hb-         I2— 

„„    3. 1416-3- 160  ,a  .    ...    ,.      v 

RS  = rx (Substitution.) 

72/S=  125.664  ft,  per  min. 

Example  2.  At  how  many  revolutions  per  minute 
must  a  4-in.  bolt  be  turned  to  give  a  cutting  speed  of 
60  ft.  per  min.? 

Solution. 

7rrf(R.P.M.) 


RS  = 


12 


_     3. 1416-4- (R. P.M.)        ,a  .    ...    ..      v 

60  = — r^ (Substitution.) 

60- 12  =  3. 1416-4-  (R.P.M.)      (Clearing  of  fractions.) 
_60  12       RpM       (Dividing  by  the  coefficient  of  R.P.M.) 
R.P.M.  =  57. 6+in. 


RIM   SPEED  81 

Example  3.  A  pulley  on  a  shaft  turning  at  140  R.P.M. 
is  to  furnish  a  rim  speed  of  1500  ft.  per  min.  Find  the 
diameter  of  the  pulley  that  must  be  placed  on  the  shaft. 

Solution. 

xrf(R.P.M.) 
RS= 12 

ir  M    3.1416-cM40  ,a  ,    ...    ,.      x 

15.00  = r^ —  (Substitution.) 

1500 -12  =  3. 1416 •  d- 140  (Clearing  of  fractions.) 

=  d  (Dividing  by  the  coefficient  of  d.) 


3.1416-140 

rf=40.9  +  inches 

Note.  In  rough  work  where  the  exact  answer  is  not  required  the 
following  formula  is  used: 

„  _      rf(R.P.M.)  3.1416  .  ...  .  .     , 

R.S.  =  — — -,     since     — — — is  approximately  equal  to  j. 

■     4  i.1 

EXERCISE    1 

Find  the  exact  results  in  the  following: 

1.  Find  the  cutting  speed  of  a  5-in.  cylinder  being 
turned  at  75  R.P.M.  Ans.  98.175  ft.  per  min. 

2.  The  armature  of  a  dynamo  is  16  ins.  in  diameter 
and  runs  at  1350  R.P.M.     Find  the  rim  speed. 

Ans.  5654.9  ft.  per  min. 

3.  At  how  many  R.P.M.  should  a  6-in.  cylinder  be  turned 
to  give  a  cutting  speed  of  60  ft.  per  minute?        Ans.  38.2. 

4.  An  8-in.  emery  wheel  has  a  rim  speed  of  4000  ft. 
per  minute.     How  many  R.P.M.  does  it  make.    Ans.  1909. 

5.  A  shaft  is  running  at  175  R.P.M.  How  large  a  pulley 
must  be  placed  on  this  shaft  to  give  a  rim  speed  of  2250  ft. 
per  minute?  Ans.  49.11  ins. 


82 


CUTTING  SPEED,  PULLEYS  AND  GEARS 


6.  A  30-in.  pulley  runs  at  250  R.P.M.  Find  the  rim 
speed.  Ans.  1963.5. 

7.  The  pulley  of  Problem  6  is  belted  to  a  15-in.  pulley. 
What  is  the  speed  of  the  belt?  What  is  the  rim  speed  of 
the  15-in.  pulley?  Ans.  1963.5;  1963.5. 

8.  Find  the  R.P.M.  of  the  15-in.  pulley.  Ans.  500. 


7  and  8  above  illus- 
is  belted  to  a  15-in. 


70.  Pulley  Speeds.  Problems  6, 
trate  the  fact  that  if  a  30-in.  pulley 
pulley  the  R.P.M.  of 
the  15-in.  pulley  will  be 
twice  the  R.P.M.  of  the 
30-in.  pulley.  Or,  in 
general : 

Rule.  When  two  pul- 
leys are  belted  together  the 
R.P.M.  vary  inversely  as 
the  size  of  the  pulleys. 

Example    1.     A   20-in.    pulley   running   at    180   R.P.M. 
drives  an  8-in.  pulley.     Find  the  R.P.M.  of  the  8-in.  pulley. 

Solution.     Let  z  =  the  R.P.M.  of  the  8-in.  pulley.     Then 
since  180  =  R.P.M.  of  the  20-in  pulley, 


600  R.P.M. 


250  R.P.M. 


Fig.  48. 


X 
180: 


20 


8z  =  3600 
rr  =  450  R.P.M. 


EXERCISE   2 

1.  A  32-in.  pulley  running  at  150  R.P.M.  drives  a  22-in. 
pulley.     Find  the  R.P.M.  of  the  22-in.  pulley.  Ans.  218  +  . 

2.  A  30-in.   pulley  is  to  drive  a   12-in.   pulley  at  800 
R.P.M.     Find  the  R.P.M.  of  the  30-in.  pulley.   Ans.  320. 


GEARS  83 

3.  A  22-in.  pulley  on  a  shaft  running  at  264  R.P.M.  is 
to  drive  a  machine  at  570  R.P.M.  Find  the  size  of  the 
pulley  on  the  machine.  Ans.  10. 17. 

4.  A  line  shaft  running  at  150  R.P.M.  is  to  drive  a 
machine  having  a  14-in.  pulley  at  375  R.P.M.  Find  the 
size  pulley  that  will  be  required  on  the  shaft.   Ans.  35  ins. 

5.  An  electric  motor  running  at  1250  R.P.M.  and  having 
a  16-in.  pulley  is  to  drive  a  line  shaft  at  175  R.P.M.  Find 
the  size  of  the  pulley  on  the  line  shaft.        Ans.  114.3  ins. 

6.  A  30-in.  pulley  running  at  240  R.P.M.  drives  a  24-in. 
pulley.     Find  the  R.P.M.  of  the  24-in.  pulley.      Ans.  300. 

71.  Gears.  In  machines  where  driving  is  done  by 
means  of  gears  it  will  be  seen 
that,  if  two  gears  are  meshed 
together,  the  smaller  gear  will 
have  the  greater  R.P.M.  Sizes 
of  gears  are  measured  by  the 
number  of  teeth. 

Rule.     When     two    gears    run 
Fig.  49.— Gears  in  Mesh,      together,  the  R.P.M.  varies  invers- 
ely as  the  number  of  teeth. 

EXERCISE  3 

1.  A  48-tooth  gear  is  driving  a  72-tooth  gear.  Find 
the  R.P.M.  of  the  72-tooth  gear  if  the  48-tooth  gear  is 
running  160  R.P.M.  Ans.  106.7. 

2.  A  72-tooth  gear  running  at  190  R.P.M.  is  to  drive 
another  gear  at  360  R.P.M.  Find  the  number  of  teeth  in 
the  second  gear.  Ans.  38. 

3.  A  26-tooth  gear  running  at  105  R.P.M.  is  to  drive  a 
14-tooth  gear.     Find  the  R.P.M.  of  the  14- tooth  gear. 

Ans.  195. 


84 


CUTTING   SPEED,   PULLEYS  AND   GEARS 


EXERCISE  4 
(Miscellaneous  Problems) 

1.  A  locomotive  has  a  6-foot  drive  wheel.  Find  the 
R.P.M.  of  the  wheel  when  the  engine  is  running  at  50  miles 
per  hour.  Ans.  700.2. 

2.  Cone  pulleys: 


1050  R.P.M. 


Fig.  50. 


From  Fig.  50  find  the  three  speeds  of  the  lower  pulley. 

Ans.  600,  1260,  2800. 
3.  Pulleys: 


Fig.  51. 


Pulley   1   is  a  16-in.   pulley  and  runs  at   125   R.P.M. 
Pulley  2  is  a  10-in.  pulley. 


REVIEW    PROBLEMS 


85 


Pulley  3  is  an  18-in.  pulley. 

Pulley  4  is  an  8-in.  pulley. 

Pulleys  2  and  3  are  on  the  same  shaft. 

Find  the  R.P.M.  of  pulley  4. 

4.  Gears: 


Ans.  450. 


Gear  1  has  42  teeth  and  should  be  run  at  800  R.P.M. 
Gear  2  has  96  teeth. 
Gear  3  has  48  teeth. 

Gear  4  is  on  a  shaft  running  at  300  R.P.M.     How  many 
teeth  has  gear  4?  Ans.  56. 

5.  Dynamo  under  a  passenger  car: 


Fig.  53. 

Circle  1  represents  a  40-in.  car  wheel. 

Circle  2  represents  a  14-in.  pulley  on  the  axle  of  the  car. 

Circle  3  represents  a  6-in.  pulle3r  on  a  dynamo  and  is 
driven  by  a  belt  from  the  pulley  on  the  car  axle. 

Find  the  R.P.M.  of  the  dynamo  when  the  car  is  running 
at  30  miles  per  hour.  Ans.  588. 


CHAPTER  VII 

ELECTRICAL    FORMULAS    INVOLVING    SQUARES  AND 
SQUARE   ROOTS 


72.  Power  in  a   Direct  Current  Circuit.     In   a  circuit 
carrying  direct  current, 

P=EI=PR=^,    E  =  IR 

K 


where,  P  =  power  in  watts; 

E  =  EM.F. 

in  volts; 

I  =  current 

in  amper 

es; 

R  =  resistance  in  ohms. 

EXERCISE   1 

Find  the 

missing  terms: 

P 

E 

I 

R 

Ans. 

1. 

110 

.85 

93.5, 

129.41 

2. 

220 

.425 

93.5, 

517.64 

3. 

1.2 

220 

316.8, 

264. 

4. 

110 

225 

53 .  77, 

488. 

5. 

220 

32 

1512.5 

,6.875. 

6. 

.55 

200 

60.5, 

110. 

7.     40 

.38 

105.26 

,276.3. 

8.     75 

240 

134.2, 

558. 

9.     28. 

125 

80 

47.43, 

592. 

10.    100 

55 

1.818, 

30.25. 

86 


DIRECT  CURRENT  CIRCUIT  87 

11.  Three  resistances  of  220,  234  and  431  ohms  respect- 
ively are  connected  in  series  and  a  pressure  of  550  volts 
is  applied.     How  much  power  is  used?  Ans.  341.8  watts. 

12.  If  the  above  resistances  are  connected  in  parallel, 
how  many  volts  must  be  applied  to  get  the  same  power? 

Ans.   175.19. 

13.  Which  uses  the  more  power,  a  lamp  with  a  resist- 
ance of  220  ohms  on  a  110  volt  circuit  or  lamp  on  a  110 
volt  circuit  that  uses  2 .  24  amperes?  Ans.  Second  one. 

14.  A  110  volt  arc  light  requires  12  amperes  to  operate. 
How  many  watts  are  used?     What  is  the  resistance? 

Ans.   1320  watts,  9.17  ohms. 

15.  An  arc  light  requires  20  amperes  to  operate  at  45 
volts.  How  many  horse-power  does  it  take?  (1  horse 
power  =  746  watts.)  Ans.   1.206. 

16.  A  12  candle  power  lamp  on  a  110  volt  circuit  takes 
0.25  ampere.     How  many  watts  per  candle  power  are  used? 

Ans.  2.29. 

17.  A  generator  is  producing  30,000  watts  at  225  volts. 
What  is  the  current  flowing?  Ans.   133.3. 

73.  Force  Between  Two  Magnets.  The  force  between 
two  magnet  poles  is  expressed  by  the  formula: 

MiM2 
*-     D2    > 

where,       F  =  force  in  dynes  between  two  magnets; 
M i  and  M2  =  pole  strengths  of  the  two  magnets  in  unit  poles; 
D  =  distance  between  the  two  magnet  poles  in  centi- 
meters, (1  inch  =  2. 54  centimeters). 


88  ELECTRICAL  FORMULAS 


EXERCISE   2 

rin< 

1  the  missing  terms: 

F 

Mi 

M2 

D 

1. 

2 

8 

4 

Ans. 

1. 

2. 

4 

12.5 

8 

5. 

3. 

12 

3 

2 

16. 

4. 

9 

7 

2 

15.75 

5. 

10 

10 

12 

144. 

6. 

20 

20 

16 

4. 

7. 

4 

6 

1 

24. 

8.  Change  to  centimeters:  3",  4",  12". 

Ans.  7.62,  10.14,  30.45. 

9.  Change  to  inches:    10  cm.,  50  cm.,  75  cm. 

Ans.  3.94",  19.7",  29.55". 
10.  If  a  magnet  pob  of  10  unit  poles  is  placed  4  cm. 
from  another  magnet  of  24  units  pole  strength,  what  force 
will  they  exert  upon  each  other?  Ans.    15  dynes. 

74.  Horse-power    of    an    Electric    Motor.     The    horse- 
power of  a  motor  is  expressed  by  the  formula: 

2*nT 
33000' 

where,   H.P.  =  horse  power; 
7T  =  3.1410; 

n  =  R.P.M.  of  the  motor; 
T  —  torque  in  lbs.  ft. 


HORSE-POWER  OF  AN  ELECTRIC  MOTOR  89 


Six-Pole  Direct  Current  Generator. 


Armature  of  a  Direct  Current  Generator. 


90  ELECTRICAL  FORMULAS 

EXERCISE  3 

Find  the  missing  terms: 


H.P. 

n 

T 

Ans. 

1. 

1500 

18.4 

5.255, 

2. 

5 

1800 

14.6. 

3. 

1.5 

1500 

5.25. 

4. 

10 

39.6 

1328. 

5. 

8.5 

1650 

27.1. 

6. 

6.454 

1200 

28.25. 

7.  A  motor  running  at  1040  R.P.M.  develops  a  torque 
of  931  lbs.  ft.     Find  the  horse  power?  Ans.   185. 

8.  A  motor  is  to  run  at  1680  R.P.M.  If  it  has  a  12" 
pulley  and  is  to  furnish  a  pull  of  200  pounds  on  the  belt, 
what  horse  power  will  be  required?  Ans.  32. 

9.  A  10  H.P.  motor  with  an  8"  pulley  is  to  drive  a  30" 
pulley  at  480  R.P.M.     Find  the  torque.  Ans.  29. 18. 

10.  An  80  horse  power  motor  developing  a  torque  of 
1231  lbs.  ft.  and  having  a  10"  pulley  drives  a  48"  pulley. 
Find  the  R.P.M.  of  the  48"  pulley.  Ans.  71. 

11.  An  electric  car  has  a  gear  ratio  of  14  teeth  on  the 
motor  sprocket  to  65  teeth  on  the  axle  sprocket.  The 
wheels  are  33".  The  car  has  a  20  H.P.  motor  which  gives 
a  torque  of  84.50  lbs.  ft.  Find  the  maximum  speed  of  the 
car  if  the  motor  is  75  per  cent  efficient.  Ans.   19 . 7. 

75.  Voltage  of  a  D.  C.  Generator.  Voltage  for  direct 
current  generators  is  expressed  by  the  formula: 

F=  N<f>Zp 
10V60' 


MURRAY  LOOP  FORMULA 


91 


where,  #  =  E.M.F.  in  volts; 

Z  =  number  of  active  conductors  on  the  armature; 
<{>  =  flux  in  maxwells; 
p  =  number  of  poles; 
n  =  number  of  revolutions  per  minute; 
p'  =  number  of  brush  arms. 


EXERCISE  4 

Fi 

nd  the 

missing 

numbers : 

E 

Z 

<t> 

n     p 

V' 

Ans. 

1. 

200 

1000000 

100  2 

2 

70. 

2. 

220 

226 

1500  2 

2 

3020000. 

3. 

116 

24 

2400  8 

2 

4840000, 

4. 

120 

200 

1671000 

2 

2 

2160. 

5. 

1040 

2400 

2600000 

1200 

2 

2. 

6. 

112 

400000 

2400  4 

4 

700. 

7. 

1040 

260 

4000000 

1500  8 

ft 

2. 

8.  A  generator  has  a  speed  of  1980  R.P.M.,  an  E.M.F. 
of  50  volts,  4  poles,  200  turns,  and  4  brushes.  What  will 
be  the  flux?  Ans.   1515000. 

9.  How  many  poles  will  be  required  in  a  generator  for 
a  voltage  of  520,  2000  surface  conductors,  flux  of  1300000 
maxwells  and  a  speed  of  1200  R.P.M.  and  4  brush  arms? 

Ans.  4. 

76.  Murray  Loop  Formula.     The  formula 


X 


R2  = 

Ri    L-X' 


is  used  to  find  the  distance  from  a  station  along  a  telephone 


92  ELECTRICAL  FORMULAS 

line  to  the  place  where  a  wire  is  grounded;   thus  in  Fig.  54, 


X 
Fig.  54. 

Station 

Ground 

Ri  and  Ro  are  resistances  in  a  bridge  at  the  station: 
L  =  total  length  of  the  line  and  return ; 
X  =  distance  from  the  station  to  the  place  where  the 
wire  is  grounded. 


EXERCISE  5 

Find  the  missing  terms: 

Ri 

R-2 

L 

X 

1.          75 

50 

25  miles 

Ans.   10. 

2.          84 

20 

163  miles 

31.35. 

3.        128 

8 

10  miles 

.588 

4.  Find  the  formula  for  X  in  terms  of  Ri,  Ro,  and  L. 

.         „       RoL 

*"■■  X=5T+B2- 

77.  Designing  an  Inductance  Coil.     The  formula 
_  4iT2n2r2u  1 .  26  n2uA 

wh~   or     ~m    ' 

is  given  for  finding  the  inductance  of  a  coil; 

L  =  inductance  in  henries; 

n  =  number  of  turns; 

r  =  radius  of  the  coil  in  centimeters; 

,  ...         .    .  (  =1  for  air 

u  —  permeability  ot  the  core  \       -,r™  r 

i    =  loOO  for  iron. 

A  =  cross  section  area  of  the  core. 


DESIGNING  A  CONDENSER  93 


EXERCISE  6 


1.  Find  the  inductance  of  the  primary  coil  of  a  trans- 
former having  400  turns  if  the  iron  core  has  a  cross  section 
area  of  300  sq.  cm.  and  is  60  cm.  long. 

Ans.   15 . 1  henries. 

2.  Design  the  dimensions  of  an  inductance  coil  wound 
on  an  iron  core  that  will  have  an  inductance  of  25  henries 
and  be  of  good  shape. 

78.  Designing  a  Condenser.  The  following  formula  is 
given  for  the  capacity  of  a  condenser 

_885Xa(ft-l) 
tmf~~~W"d        ' 

Cmf=  capacity  of  a  condenser  in  microfarads; 
K  —  dielectric  constant ; 
a  =  area  of  one  plate; 
n  =  number  of  metal  plates ; 
d  =  thickness  of  dielectric  plate  in  centimeters. 

EXERCISE  7 

1.  An  aluminum  and  air  condenser  has  25  aluminum 
plates  of  diameter  12  cm.,  separated  by  2  mm.  of  air.  Find 
the  capacity  if  K  =  1  for  air.  Ans.    .0012. 

2.  A  condenser  is  to  be  made  of  mica  and  tinfoil  to 
have  a  capacity  of  .32  microfarad.  The  sheets  are  20  cm. 
square  and  the  mica  is  .05  cm.  thick,  K  for  mica  =  6.  How 
many  sheets  are  needed?  Ans.  76. 

3.  Design  the  dimensions  of  a  condenser  of  tinfoil  and 
glass  that  will  have  a  capacity  of  0.002  microfarad.  Assume 
that  the  glass  available  has  a  thickness  of  0.25  cm. 

K  =  6  for  glass. 


94  ELECTRICAL  FORMULAS 

EXERCISE  8 

General  Formulas.     Evaluate  the  following  formulas: 

-  -b-Vb2+4ac        .  ,    ,      . 

1.  x  =  —     — ^ —     — ,     when     a=l,b  =  4,c  =  5. 

Ans.   1. 

-b-Vb2+4ac         ,  ,    ,      .  B 

2.  x=—     — - —     — ,     when    a  =  l,  6=4,  c  =  5. 

Ans.    —5. 

3.  x  =  —     — ~—     — ,     when     a  =  3,  b=  —  14,  c=  —  6. 

Ans.  4.1893  and  0.4774. 

4.  6  =  Va2+c2-2a'c,     when     a  =  7,  c  =  5,  a'  =  3. 

Ans.  6.633. 

5.  A  =  Vs(s  -  a  )  (s-b)  (s-c) ,     when     a  =  16,  b  =  12, 

c=20  and  s=§(a+6+c).  Ans.  96. 

6.  A  =  Vs(s-a)  (s-6)  (s-c),     when     a  =  22,  6  =  30, 

c=26ands  =  |(a+&+c).  Ans.  278.5. 


CHAPTER  VIII 
QUADRATIC  EQUATIONS 

79.  Definitions.  Some  formulas  and  problems  require 
the  use  of  an  unknown  letter  raised  to  the  second  power, 
giving  an  equation  containing  x2  or  y2,  etc.  An  equation 
that  contains  an  unknown  in  the  second  power  as  the  highest 
power  of  the  unknown  is  a  quadratic  equation. 

If  a  quadratic  equation  contains  only  the  second  power, 
of  the  unknown,  it  can  be  put  in  the  form  x2  =  25.  (Axiom : 
If  two  expressions  are  equal  their  square  roots  are  equal.) 

Therefore,  since 

x2=     25 
x=zk  5  (±  is  read  +  or  — ) 

Observe  that: 

1.  The  sign  ±  means  that  both  +5  and  —5  check  the 
equation. 

2.  The  square  root  of  a  number  may  be  positive  or  negative 
because  (  +  5)2  =  25  and  (-5)2  =  25. 

3.  A  quadratic  equation  has  two  answers. 


EXERCISE    1 

Solve  the  following  equations: 

1.     z2=121. 

Ans.    £=±11. 

2.     x2  =  50. 

£=±7.071. 

3.    R2  =  l 

#=±.866. 

4.     d2  =  U. 

d=±%. 

9G 


QUADRATIC   EQUATIONS 


y     8- 


6.  ?/2-12  =  37. 

7.  5a;2  =180. 

8.  3a-2 +13  =  1  GO. 


Ans.  y=  ±935. 
I/=±  7. 
z=±  0. 
x  =  ±     7. 


80.  Quadratic  Equations  with  Both  First  and  Second 
Powers.  A  quadratic  equation  which  contains  the  first 
and  second  powers  of  the  unknown  cannot  be  solved  by  the 
above  method.  For  example,  in  x2-{-6x  =  40,  the  square 
root  of  x2-\-6x  cannot  be  found.  The  form  of  the  equation 
can  be  changed  so  that  the  square  roots  of  both  members 
can  be  found. 


EXERCISE   2 


Expand  the  following: 

1.  (x+1)2.  Ans.  x2  4-2x4-1. 

2.  0  +  2)2 

3.  (x-1)2 

4.  (x-4)2 

5.  0  +  5)2 

6.  (x  +  i)2 

7.  Or  +  f)2 

8.  (x-h)2 

Observe  that: 

1.  The  aiibivers  to  all  the  'problems  in  Exercise  2  contain 
three  terms.     An  x2  term,  an  x  term,  and  a  numerical  term. 

2.  The  x2  term  and  the  numerical  term  are  perfect  squares 
and  the  x  term  is  twice  the  product  of  the  two  terms  of  the 
problem.  (For  example,  in  (x  —  4)2  =  x2  —  8x-\-lQ,  x2  and 
16  are  the  squares  of  x  and  4,  while  —8x  is  2x(  —  4).) 


TRINOMIAL  SQUARES  97 

81.  Trinomial  Squares.  An  expression  having  two 
terms  which  aie  perfect  squares  and  the  other  term  twice 
the  product  of  the  square  roots  of  these  two  terms  is  a 
trinomial  square.  The  square  root  of  a  trinomial  square  can 
be  found.  The  process  is  the  reverse  of  the  method  used  in 
Exercise  2. 

Rule.  To  find  the  square  root  of  a  trinomial  square  take 
the  square  root  of  the  two  -perfect  square  terms  and  connect 
them  by  the  sign  of  the  other  term. 

Example  1.     V~i2+  Qx+  9=±(z+3). 

Example  2.     Vx2 -  10a; +25  =  ± (x- 5). 

EXERCISE   3 

Find  the  square  root  of  the  following;  give  positive 
results  only: 

1.  z2+2a;+l.  Ans.  x+1. 

2.  :r2-4x+4.  x-2. 

3.  x2-Gx+9. 

4.  x2+14x+49. 

5.  ?/2-l  6?/ +  64. 

6.  x2+20x+100. 

7.  R2-R  +  l 

4        4 

8.  a2  —  ^a-\-q. 

82.  To  Complete  a  Trinomial  Square.  A  quadratic 
equation  of  the  form  z2  +  6a:  =  40  can  be  changed  to  an 
equation  having  a  trinomial  square  in  the  first  member 
by  adding  9  to  both  members,  thus: 

x2  +  Sx       =40 

z2+6z+9  =  49 


98  QUADRATIC  EQUATIONS 

The  number  9  which  must  be  added  to  z2+6;r  to  make 
it  a  trinomial  square  is  the  square  of  \  of  6  or  9.  (This 
follows  from  Exercises  2  and  3.) 

Rule.  To  change  an  expression  of  the  form  x2-\-ax  into 
a  trinomial  square  add  to  it  the  square  of  \  the  coefficient  of  x. 

Example  1.  Find  the  number  which  will  make  x2-\-\()x 
a  trinomial  square. 

Solution.  The  coefficient  of  x  is  10,  §  of  10  =  5,  52  =  25. 
Therefore  25  added  to  x2+lQx  will  make  it  a  trinomial 
square. 

Example  2.  Find  the  number  which  will  make  x2 -7x 
a  trinomial  square. 

Solution  The  coefficient  of  x  is  7,  \  of  7  =  $,  (1)2  =  Y- 
Therefore  ^  added  to  x2  -  Ix  will  make  it  a  trinomial  square. 


EXERCISE   4 

Find  the  number  which 

will 

add  t 

o  the  following  to  make 

rinomial  squares: 

1.  x2  +  2x. 

Ans. 

1. 

2.  x2-(jx. 

9. 

3.  x2+U)x. 

25. 

4.  x2  +  5x. 

25 
4~- 

5.  x2  —  9x. 

8  1 

6.  x2-12z. 

36. 

83.  Solution  of  Quadratic  Equations  by  Completing  the 
Square. 

Example  1.     :r2  +  6a;  =  40 

af'-f  6:c+9  =  49      (Adding  9  to  both  members  to 
make  the  first  member  a  tri- 
nomial square) 
z+3  =  ±7    (Extracting  the  square  root  of 
both  members) 


TRINOMIAL  SQUARES  99 

Then        z+3=+7  and  x+3=-  7 

x  =  4  x=-10 

Check.  424-6.4  =  40  (-10)2+6(-10)=40 

16+24  =  40  100-60  =  40 

40  =  40  40  =  40 

Example  2.    2x2-  Qx  =  12 4- 4x 
2x2-10.r=12 

x2_  5£=  g  (The  coefficient  of  x2  must 
be  made  equal  to  1) 
x2-hx-\-2i=^i-  (Adding  ?-£-  to  both  mem- 
bers to  make  the  first 
member  a  trinomial 
square) 


Then  »—'!=+$     and     x—%= 

-r  —  1-2  r  =  _  2 

£  =  6  z=  —  1 


Observe  that: 

1.  A  quadratic  equation  has  two  answers. 

2.  Both  answers  must  check. 

3.  Before  completing  the  square  the  first  term  must  have 
the  coefficient  1. 

EXERCISE  5 

Solve  the  following  equations: 

1.  z2+4:c  =  45.  Ans.  x  =  5  or  -9. 

2.  z2  +  6:r  =  27.  a;  =  3  or  -9. 

3.  x2-5.r  =  24.  x  =  8  or  -3. 

4.  2x2-7.r  =  30.  x  =  6  or  -2\. 

5.  2x2-7x  =  34.  z  =  6.229  or  2.729. 

6.  Zx2-\  ".r  =  8.  x=l  or  -2f. 


100 


7.  x2+x2+10x  =  22  +  3x. 
^_7z  =  51 

8-    2      8       4- 


QUADRATIC   EQUATIONS 

Ans.    x  =  2  or 


or 


-4 


84.  Solution  of  Quadratic  Equations  by  Formula.  Any 
quadratic  equation  can  be  expressed  in  the  form  ax2-\-bx  =  c, 
where  a,  b  and  c  are  general  numbers.  If  the  equation 
ax2  +  bx  =  c  is  solved  for  x  by  completing  the  square,  it  will 

be  found  thai  

-6±\/62+4ac 


x  = 


2a 


The  above  result  should  be  memorized  and  used  as  a  formula 
for  solving  other  quadratic  equations  of  the  form  ax2  +  bx  =  c. 
Observe  that  a  is  the  coefficient  of  x2,  b  is  the  coefficient 
of  x  and  c  is  the  term  that  does  not  contain  x. 


Problems. 

=  c. 

Solutions. 

1.  ax2-\-bx  = 

-b±Vb2+4ac 
X~             2a 

=  18 

a -   '2 
b=   5 
c=18 

2.  2x2  +  5x  = 

-5±V52+4-2-18 

X~               2-2 

Simplifying: 

-5±V25  +  144 
4 

-5±Vl69 
X=          4 

-5±13 

*~       4 

8             -18 
£  =  t     or     — t— 
4               4 

x  =  2    or     -4| 

SOLUTION    BY   FORMULA 


101 


Substitute  in  the  formula. 


a  =       3 

3.  3z2-4:r=15       b=-  4 

c=      15 

4.  a;2  — 2a;  — 5  =  45  — 3a\ 

a;2  — 5a:  =  50.     (Arranging  in  the  standard  form.) 
a=       1 
b=-  5 
c=     50 


Substitute  in  the  formula. 


EXERCISE    6 


Solve  bv  the  formula  method: 


1.  2x2-\-5x  =  S. 

2.  5a;2+3x  =  2. 

3.  a;2+4a;  =  5. 

4.  a-2  -2x  =15. 

5.  a;2  =  4a;+12. 

6.  2.r2-3a;=18. 

7.  7x2-4a;-8=10. 

8.  5t2-6z  =  41. 

9  ^_  =  _^. 
3x-7     a;+84 

10.  a;2  +  3a;  =  9. 

11.  3a;2  +  5x  =  2. 


Ans. 


-3,  +.5. 

I  -1. 
1,  -5. 

5,  -3. 
0,  -2. 

3.84,  -2.34. 
1.889,  -1.318. 
3.526,  -2.326. 

14,   -10. 

-4.854,  1.854. 
—  2    i 


85.  Application  of  the  Quadratic  Equation  to  the  Right 
Triangle.     If  the  hypotenuse   of  a  right  triangle  (Fig.  55) 


102 


QUADRATIC   EQUATIONS 


is  c  and  the  sides  arc  a  and  b  the  relation  between  the  sides, 
and  the  hypotenuse  is  expressed  by  the  formula, 

(?  =  a?+b2. 


EXERCISE   7 


1. 


v+L     Find  the  sides.        Ans.   12,  16. 


2x  +  a 


Find  the  sides. 

Ans.  5,  13. 


3.  One  side  of  a  right  triangle  is  2  more  than  the  other 
side  and  the  hypotenuse  is  10.     Find  the  sides.      Ans.  6,  8. 

Hint.     Let  x  =  ovc  side 
then  £+2  =  the  other  side 


4.  One  side  of  a  rectangular  lot  is  31  rods  more  than  the 
other  side  and  the  area  is  360  square  rods.  Find  the  dimen- 
sions. Ans.  9,  40. 


RIGHT  TRIANGLES 


103 


Fig.  58. 

5.  Find  the  side  of  the  square  in  Fig  58. 

Ans.  18. 

6.  Find  the  inside  and  outside  radius  of  a  ring  that  has 
an  area  of  34.5576  square  inches  and  a  thickness  of  1  inch. 

Ans.  5,  6. 


Fig.  59. 


Find  the  size  plug  that  will  fit  as  shown  in  Fig.  59. 

Ans.  Radius  =  1.0858. 


CHAPTER  IX 
SIMULTANEOUS  EQUATIONS 

86.  Definition.  In  some  problems  which  involve  two 
or  more  unknown  quantities  it  is  often  difficult  to  express 
both  unknowns  in  terms  of  one  letter.  Such  problems  can 
be  solved  conveniently  by  using  two  or  more  unknown 
letters. 

For  example,  The  sum  of  two  numbers  is  29  and  the 
difference  is  11.     Find  the  numbers. 
Let  z  =  one  of  the  numbers 

y  =  the  other  number 
Then  x+y  =  29 

x  —  y  —  11 

Neither  one  of  these  two  equations  can  be  solved  inde- 
pendently; they  must  be  solved  together.  Two  or  more 
equations  involving  two  or  more  unknowns  which  must  be 
solved  together  are  called  simultaneous  equations. 

87.  Solution  of  Simultaneous  Equations.  Two  simul- 
taneous equations  can  be  solved  by  eliminating  one  of 
the  unknowns. 

(a)  Elimination  by  Addition  or  Subtraction: 

Example  1.  The  two  simultaneous  equations  above 
can  t)o  solved  by  adding  the  first  member  of  the  one  to  the 
first  member  of  the  other,  and  the  second  member  to  second 

104 


ELIMINATION   BY   ADDITION   OR   SUBTRACTION      105 

member.     The   sums   will   be    equal.      (Axiom:     If   equals 
are  added  to  equals  the  sums  are  equal.) 

.T  +  7/  =  29         .......        (1) 

x-y^ll (2) 

2.t  =  40  Adding  (1)  and  (2)     (3) 
a;  =  20 

20  +  */=  29  Substituting  value  of  x 

V=   9  in  (1), 

Check.  20+9  =  29 

29  =  29 

20-9=11 

11  =  11 

Note.  When  one  unknown  is  found  the  other  may  be  found  by 
substituting  the  value  found  in  either  of  the  original  equations.  To 
check  simultaneous  equations  the  values  found  must  check  both  equations. 


Example  2. 


x+Zy=   8 
2x+y=l\ 

2z+6y=16 


Multiplying  both  mem- 
bers of  (1)  by  2. 


(1) 
(2) 
(3) 


-5y  =  -5. 

y=  1. 
x+3=   8. 


Subtracting  (3)  from  (2). 
Substituting  in  (1). 


Check. 


5+3  = 


10+1  =  11. 
11=11. 


106  SIMULTANEOUS   EQUATIONS 

Example  3.     7z+3?/=16 (1) 

Zx+2y=   9 (2) 

Ux+Gij  =  32.       (1)X2 (3) 

9z+6?/  =  27.       (2)X3 (4) 

5x  =   5.       (3)- (4). 
x=    1. 
7  +  3?/=  16.       Substituting  in  (1), 
3?/=   9. 
2/=3. 

CTiecfc.  7+9=16. 

16=16. 

3  +  6=   9. 

9=   9. 

Observe  that: 

1.  Either  unknown  may  be  eliminated. 

2.  To  eliminate  an  unknown  it  may  be  necessary  to  mul- 
tiply both  members  of  one  equation  by  some  number,  and  in 
some  problems  it  may  be  necessary  to  multiply  both  members 
of  both  equations  by  some  number.  Both  equations  do  not  have 
to  be  multiplied  by  the  same  number. 

3.  In  checking  simultaneous  equations  the  values  found 
must  check  both  equations. 


EXERCISE    1 

Solve  the  following  simultaneous  equations: 

1.  2x+3i/=16. 

2x-   y  =   8.  Ans.  x  =  5,  y  =  2. 

2.  2x+  y=   4. 

3s-  ?/ =  21.  x=5,  y=-Q. 

3.  &r+  y=   7. 

llx+2?/  =  28.  x=-2$,  y  =  29l 


ELIMINATION  BY  SUBSTITUTION 


10, 


4.  x-2y=-12. 

Ax—  y  =    1. 

5.  3z  +  5?/  =  33. 
4x-2y=18. 


Ans.  x  =  2,  y-7. 
x  =  Q,  ?/  =  3. 


(6)  Elimination  by  Substitution.  Some  simultaneous 
equations  can  be  solved  more  efficiently  by  another  method 
called  substitution.  In  this  method,  solve  either  equation 
for  either  unknown  in  terms  of  the  other  unknown  (choose 
the  one  giving  the  simplest  solution),  and  substitute  the  value 
found  in  the  other  equation. 


Example  1.        x+y  =  5 

2x+3i/=13 

x  =  5  —  y      Solving  (1)  for  a;. 

2(5-y)+Sy=  13      Substituting  in  (2), 

10-2y+3y=13 

y=  3 

z+3  =   5 

x=   2 

Check.  2+3=   5 

5=   5 

2-2+3-3  =  13 

4+9  =  13 

13=13 


(1) 
(2) 
(3) 


Example  2. 


3x+2y  =  25      . 
2x  —  5y=   4 

2x  =   4  +  5?/ 
4+5?/ 


(1) 
(2) 


From  (2), 


x  = 


3(4+57/) 


+2?/  =  25     Substituting  in  (1), 


3(4  +  57/)+4?7  =  50 


108  SIMULTANEOUS  EQUATIONS 

12+15*/+4y  =  50 
19y  =  38 

y=  2 

2x-10=  4 

2x=14 
x=   7 

EXERCISE   2 

Solve  by  the  substitution  method: 
1.  2z+3?/  =  26. 


x  —  5y  =   0. 

Ans. 

z=10,  y=2. 

2. 

3i/= 5s- 18. 

3y 

a:  =  6,  y  =  4. 

3. 

as=2y+3. 
z=7y-12. 

• 

z  =  9,  ?/  =  3. 

4. 

?/-3z=2. 

£2+y2  =  4. 

x  =  0       rr=-l|. 

O    01'                       13 

«/=2       2/=-l|. 

5. 

4^+2/2  =  5. 

y=2°Ty=-2. 

CHAPTER  X 

THE  GRAPH 

88.  Plotting  a  Graph.  The  number  of  revolutions  per 
minute,  that  will  give  a  cutting  speed  of  45  ft.  per  minute 
to  the  following  size  stock,  may  be  found  and  the  results 
expressed  in  tabular  form  thus: 


R.P.M. 


Diameter 
of  Stock. 

1.  \"  343 

2.  |"  229 

3.  1"  172 

4.  \\"  137 

5.  \\"  114 

6.  2"  86 

7.  2§"  69 

8.  3"  57 

These  results  may  also  be  represented  graphically  by 
plotting  the  diameters  along  a  horizontal  line  (called  an 
axis)  and  the  R.P.M.  along  a  line  perpendicular  to  this 
line  (also  an  axis),  as  in  Fig.  64. 

In  Fig.  64  point  1  corresponds  to  a  diameter  of  \  in. 
and  343  R.P.M.     Similarly  point  2  may  be  located,  etc. 

Observe  that: 

1.  All  of  the  points  are  connected  by  a  smooth  regular 
curve. 

109 


110 


THE   GRAPH 


2.  All  points  on  the  curve  represent  some  corresponding 
diameter  and  R.P.M.  that  will  give  a  cutting  speed  of  45  ft. 
per  minute. 

3.  When  either  diameter  of  stock  or  R.P.M.  is  given,  the 
other  can  be  found  from  the  figure. 

The  horizontal  axis  is  called  the  X-axis  and  the  vertical 


l  2 

Diameter  of  Stock 

Fig.  60. 


axis  is  called  the  Y-axis.  The  curve  connecting  the  points 
located  is  a  graph. 

Example.  -From  Fig.  60,  find  the  R.P.M.  necessary  to 
give  a  2;j-in.  stock  a  cutting  speed  of  45  ft.  per  minute. 

Solution.  Find  2 \  in.  on  the  X-axis.  Find  the  point 
on  the  graph  directly  above  this  point  and  read  the  R.P.M. 
on  the  Y-axis  (75)  corresponding  to  this  point. 


CUTTING  SPEED  111 


EXERCISE   1 

From  Fig.  60  find  the  R.P.M.  necessary  to  give  a  cutting 
speed  of  45  ft.  per  minute  to  the  following  stock : 

Diameter  R  p  M 

of  Stock. 

1.  2f" 

2.  2*" 


3. 

4. 


■8 

1" 

8 


EXERCISE  2 


1.  Make  a  table  of  diameters  and  R.P.M.  as  on  page  109, 
for  a  cutting  speed  of  85  ft.  per  minute.  (The  slide  rule 
can  be  used  to  advantage  here.) 

2.  Plot  on  squared  paper  a  graph  of  the  data  computed 
in  Problem  1. 

Note.  The  value  of  the  units  on  the  squared  paper  must  be 
so  chosen  that  the  curve  will  all  lie  on  the  sheet  of  paper,  and  the  units 
must  be  sufficiently  large  to  permit  reading  nearly  accurate  values 
from  the  curve.     (The  larger  the  curve  the  more  accurate  the  results.) 

3.  From  the  curve  of  Problem  2,  find  the  R.P.M.  neces- 
sary to  give  the  following  stock  a  cutting  speed  of  85  ft. 

per  minute: 

Diam.  R.P.M. 

(a)  I" 

(b)  If" 

(c)  2f  " 

(d)  2} " 

4.  Students  having  use  in  the  shop  for  a  set  of  these 
curves  should  plot  a  curve  for  several  other  cutting  speeds 
in  common  use.  These  curves  can  all  be  plotted  on  the 
same  scale  and  axes. 


112  THE  GRAPH 

EXERCISE  3 

1.  A  storage  battery  is  discharged  in  100  minutes  and  the 
voltage  furnished  by  the  battery  measured  at  different 
times  during  the  discharge.  Following  is  a  table  of  the 
data  read: 


Voltage. 


Time 
(in  minutes). 

0  6 


2  5.9 

4  5.8 

10  5.7 

12  5.7 

17  5.6 

22  5.5 

27  5.5 

32  5.5 

37  5.4 

42  5.3 

52  5 

62  4.7 

72  4.3 

82  3.7 

92  2.7 

94  2.5 

97  1.9 

100  1.3 

(a)  Plot  a  graph  of  the  discharge,  laying  off  the  time  on 

the  X-axis  and  the  voltage  on  the  Y-axis.  Connect  the 
points  plotted  by  a  smooth  curve. 

(6)  Read  from  the  curve  the  voltage  at  the  end  of  30 
minutes. 

(c)  What  does  the  shape  of  the  curve  indicate? 


COORDINATES 


113 


EXERCISE   4 


1.  Compute   the   weight    per   foot   of   round   steel,    and 
fill  in  the  following  table  (1  cis.  in.  of  steel  weighs  .28  lb.): 


Diam.  (Inches).                    We 

1. 

l 
8 

2. 

1 
4 

3. 

3 
8 

4. 

1 
2 

5. 

3 
4 

6. 

1 

7. 

u 

8. 

n 

9. 

2 

10. 

2\ 

11. 

3 

2.  Plot  a  curve  from  the  above  data.     Read  from  the 
curve  the  missing  values  in  the  following  table: 


Diameter. 

Weight 

(a) 

7 
8 

(b) 

13.4 

(c) 

2| 

89.  Coordinates.  Many  problems  arise  in  which  it  is 
necessary  to  plot  or  locate  points  with  reference  to  one  • 
another.  A  point  is  generally  located,  as  in  the  previous 
exercises,  by  giving  its  distance  from  each  of  two  perpen- 
dicular lines.  Such  a  system  is  a  rectangular  coordinate 
system.  The  two  distances  are  the  coordinates  of  the 
point.  The  two  perpendicular  lines  are  axes:  the  horizontal 
line  is  the  X-axis,  and  the  vertical  line  is  the  Y-axis.     The 


Ill 


THE   GRAPH 


point   where   the   axes  meet  is  the   origin.     The  axes  are 

drawn  and  lettered  as  in  Fig.  61. 

The  horizontal 
distance  of  a  point 
from  the  origin  is 
the  abscissa  of  the 
point.       The    vertical 

X1-; X     distance    of    a    point 

from  the  origin  is  the 
ordinate  of  the  point. 
The  abscissa  is  usually 
represented  by  x  and 
the  ordinate  by  ?/. 


Y' 


Fig.  61. 


90.  Location  of  Points.  Any  point  is  represented  by 
the  symbol  (x,  y)  and  a  particular  point  is  represented 
by  the  symbol  (2,  7),  (—3,  5),  etc.,  the  abscissa  always 
being  given  first.  Negative  abscissas  are  plotted  to  the 
left  of  the  origin  and  negative  ordinates  below  the  origin. 

To  locate  a  point,  lay  off 
the  axes  (squared  paper  is  the 
most  convenient  to  use) ;  then 
lay  off  the  abscissa  along  the 
X-:ixis,  and  lay  off  the  ordi- 
nate perpendicular  to  the  . 
X-axis  at  this  point. 

Example  1.  Locate  the 
point  (3,  5). 

Solution.  Measure  3  from 
the  origin  along  the  X-axis 
to    the    right,    then    5    above 

this   point.      This    is    the     point     (3,     5),     as  shown    in 
Fig.  62. 


Y' 
Fig.  62. 


GRAPH  OF  AN   EQUATION 


115 


Example  2.     Locate  the  point  (4,  -5). 
Solution.     Measure  4  from  the  origin  along  the  X-axis 
to  the  right  then  5  below  this  point,  as  in  Fig.  63. 


EXERCISE   5 

Locate  the  following  points: 

1. 

(5,  3). 

2. 

(-4,2). 

3. 

(2,  7). 

4. 

(-5,  -2). 

5. 

(4,  -3). 

6. 

(-4,  0). 

7. 

(0,  4). 

x- 


8.   (3i,  4f). 


Y' 

Fig.  63. 


91.  Graph  of  an  Equation.  An  equation  containing 
two  unknowns,  as  x-\-y  =  7,  cannot  be  solved  for  a  value  of 
x  and  y,  since  many  different  pairs  of  values  would  check, 
as  x  =  3,  y  =  4,  or  .r  =  2,  y  =  5,  etc.  By  plotting  several 
points  representing  corresponding  values  of  x  and  y  and 
connecting  these  points,  a  line  is  obtained  which  expresses 
the  relation  between  x  and  y  in  the  equation.  The  line  is 
the  graph  of  the  equation. 

Example    1.     Construct    the    graph    of    the    equation 


v- 


2x  -3. 


Solution.  Find  several  pairs  of  corresponding  values  of 
x  and  y.  This  can  be  done  most  efficiently  by  assuming 
values  of  x  and  computing  by  evaluation  the  corresponding 


116 


THE   GRAPH 


values  of  y.     For  example,  when  x  =  2,  then  ?/  =  2-2  —  3  =  1. 
Record  the  values  found  in  a  table  thus: 


Y 

/ 

0 

/ 

Y' 

X 

y 

0 

-3 

1 

-l 

2 

i 

3 

3 

4 

5 

Fig.  64. 

Plot  the  points  on  squared  paper,  as  in  Fig.  64. 

The  line  connecting  the  points  is  a  graph  of  the  equation 
y  =  2x  —  3,  and  represents  the  relation  between  x  and  y.  The 
coordinates  of  every  point  on  the  line  satisfy  the  equation, 
and  every  point  whose  coordinates  satisfy  the  equation  lies 
on  the  line.  Any  number  of  corresponding  values  of  x  and 
y  can  be  read  from  the  figure. 

EXERCISE   6 

Draw  the  graph  of  the  following  equations: 

1.  y  =  3x. 

2.  y  =  2x-l. 

3.  x-\-y  =  5.  (Solve  for  y  first.) 

4.  2x+3y  =13. 

92.  Graphs  which  Are  Not  Straight  Lines.  All  the  above 
graphs  are  straight  lines,  and  when  a  few  points  are  located 
the  whole  graph  can  be  drawn.  If  the  graph  of  an  equation 
is  not  a  straight  line,  more  points  will  have  to  be  plotted 
to  determine  the  shape  of  the  graph.  Connect  the  points 
by  a  smooth  curve.     (A  French  curve  will  assist  in  this.) 


GRAPH   OF  AN   EQUATION 


117 


EXERCISE  7 

Draw  the  graphs  of  the  following  curves: 

1.  y  =  2x2+l. 

Note.  x  =  +3  or  —3  will  give  the  same  value  of  y,  therefore  x 
has  two  values  giving  two  points  one  above  the  X-axis  and  the  other 
below  the  X-axis. 

2.  x2  +  if  =  25. 

3.  x2-y2=W. 

4.  4.r2+V  =  36. 

93.  Graphs  of  Simultaneous  Equations.     The  graph  of 
the  equation    2x  =  Sy  —  5  is  the  line 
A  B,   Fig.    (65).     The  graph  of  the 
equation  '$x-\-2y  =  12  is  the  line  CD, 
Fig.  (65). 

Observe  that: 

1.  The    coordinates   of  all  points 

on  the  line  AB  check  the  equation 

0        Q        .  Fig.  65. 

2x  —  6y  —  o. 

2.  The  coordinates  of  all  points  on  the  line  CD  check  the 
equation 


Y 

"R 

.F 

s 

Y' 

3x+2?/=12. 


3.  The  coordinates  of  the  point  P  {2,  8)  check  both  equations. 

4.  The  coordinates  of  the  point  P  are  the  same  as  the  values 
of  x  and  y  when  the  two  equations  are  solved  simultaneously. 

94.  Solution  of  Simultaneous  Equations  by  Graphs. 
If.  the  two  equations  are  graphed,  using  the  same  axes,  the 
coordinates  of  the  points  of  intersection  of  the  two  graphs 
check  both  equations  and  are  the  values  of  the  unknowns 
found  .when  the  equations  are  solved  simultaneously. 


118 


THE   GRAPH 


Example  1.  Solve  the  equations  x2+y2  =  2b  and  4x  =  3y 
by  plotting  the  graphs  and  reading  the  coordinates  of  the 
points  of  intersection  of  the  graphs,  and  check  by  solving 
as  simultaneous  equations. 

From  Fig.  66  it  will  be  seen  that  there  are  two  points  of 
intersection  P  (3,  4)  and  P'  (-3,  -4). 

Check.  :r2  +  ?/  =  25 

4x  =  3y 
x ==  ~ti 
(f2/)2+y2==25 

V 

16 
V  +  16?/2  =  400 
25?/2  =  400 
if=   16 
Z/=±4 
,    to    3(±£) 
Fig.  66.  4  4 


Y 

(T 

\j 

-'/ 

Y' 

+?/2  =   25 


EXERCISE   8 


Solve  the  following  equations  graphically  and  check  by 
solving  simultaneously : 


1.  2x+  y=Q. 

x  +  Zy=l3. 

2.  3x+2y=12. 

y  =  4x—5. 

3.  x2  +  y2  =  3G. 

x+  y=   4. 

4.  4z2+97/2  =  72. 

3y  =  2z. 


Ans.  x=l,  y  =  4. 

z  =  2,  ?/  =  3. 

s=;5.742,y=  — 1.742 
or  -x=  1.742,  ?/  =  5. 742. 

z  =  3,  ?/  =  2, 
or      x=  —3,  y=  —2. 


CHAPTER  XI 


GEOMETRY 


95.  Angles.     If  the  line  OA  (Fig.  67)  revolves  about  0 
as  center,  to  the  position  OB,  the  two  lines  form  an  angle. 
The  point  0  is  the  vertex  of  the  angle.  ^ 
The  lines  OA  and  OB  are  the  sides  of 
the  angle.     An  angle  is  read  by  reading 
the    letter  at  the  vertex  between  the 
letters  at  the  ends  of  the  sides.     Thus, 
the  angle  in  Fig.  67  is  read  angle  AOB. 
The  size  of  the  angle    is    independent 

of  the  length  of  the  sides,  and  is  measured  by  the  fractional 
part  of  a  revolution  made  in  turning  from  OA  to  OB. 

96.  Right  Angle.  If  the  line  OB  makes  one-fourth 
of  a  complete  revolution,  as  Fig.  68,  the  angle  AOB  is  a 
right  angle. 


Fig.  67. 


B 


Fig.  68. 


97.  Straight  Angle.  If  the  line  OB  makes  one-half 
of  a  complete  revolution,  as  in  Fig.  69,  the  angle  AOB  is  a 
straight  angle. 

119 


120 


GEOMETRY 


98.  Perigon.  If  the  line  OB  makes  one  complete 
revolution,  the  angle  AOB  is  a  perigon. 

99.  Degree.  A  right  angle  is  divided  into  90  equal 
parts  called  degrees.  A  degree  is  divided  into  60  equal 
parts  called  minutes.  A  minute  is  divided  into  60  equal 
parts  called  seconds.  Degrees,  minutes,  and  seconds  are 
indicated  thus:  50  degrees,  32  minutes,  24  seconds,  or 
50°  32'  24". 


100.  Protractor.     A  protractor  (Fig.  70)  is  an  instrument 
for  measuring  and  constructing  angles. 

101.  Supplementary  Angles. 
If  the  sum  of  two  angles  is 
a  straight  angle  or  180°, 
the  angles  are  supplementary 
angles.  Each  is  the  supple- 
ment of  the  other,  i.e.,  repre- 
sents what  must  be  added  to 
make  180°. 


Fig.  70. 


102.  Complementary  Angles.  If  the  sum  of  two  angles 
is  90°,  the  angles  are  complementary  angles.  Each  angle 
is  the  complement  of  the  other,  i.e.,  represents  what  must 
be  added  to  make  90°. 

103.  Angle  Theorems. 

1.  All  right  angles  are  equal. 

2.  All  straight  angles  are  equal. 

3.  Complements  of  equal  angles  are  equal. 

4.  Supplements  of  equal  angles  are  equal. 

5.  The  sum  of  the  angles  about  a  point  on  one  side  of  a 
straight  line  is  equal  to  180°. 


ANGLE   THEOREMS 


121 


6.  The  sum  of  the  angles  of  a  triangle  is  equal  to  180°. 

7.  If  two  straight  lines  intersect,  as  in  Fig.  71,  the  oppo- 
site or  vertical  angles  are  equal. 
That  is, 

Z1=Z2     and     Z3=  Z4. 

8.  If  two  angles  have  their 
sides     parallel    right     side     to 
right  side  and  left  side  to  left  side,   as  in  Fig.   72,   they 
are  equal. 


Fig.  71. 


Fig.  72. 


Fig.  73. 


9.  If  two  angles  have  their  sides  perpendicular,  right  side 
to  right  side  and  left  side  to  left  side,  as  in  Fig.  73,  they 
are  equal. 

104.  Congruent  Triangles.  Triangles  equal  in  all  re- 
spects are  congruent. 

1.  If  two  triangles  have  two  sides  and  the  included 
angle  of  the  one  equal  to  two  sides  and  the  included  angle 


Fig.  74. 


of  the  other,  the  triangles  are  congruent.  That  is,  if 
AC  =  DF,  AB  =  DE  and  Zi=ZD,  Fig.  74,  triangles  I  and 
II  are  congruent. 


122 


GEOMETRY 


2.  If  two  triangles  have  two  angles  and  the  included 
side  of  the  one  equal  to  two  angles  and  the  included  side  of 
the  other,  the  triangles  are  congruent.     That  is,  in  Fig.  75,  if 

\C  aF 


Fig.  75. 

ZA-=  ZD,  ZB=  ZE  and  AB  =  DE,  triangles  I  and  II  are 
congruent. 

3.  If  two  triangles  have  three  sides  of  the  one  equal  to 
three  sides  of  the  other,  the  triangles  are  congruent.  That 
is,  in  Fig.  76,  if  AB  =  DE,  AC  =  DF,  BC  =  EF,  triangles  I 
and  II  are  congruent. 


B     D 
Fig.  76. 


4.  Corresponding  parts  of  congruent  triangles  are  equal. 

105.  Isosceles  Triangle  Theorems. 

1.  In  an  isosceles  triangle  (Fig.  77) 
the  angles  opposite  the  equal  sides  are 
equal. 

2.  In  an  isosceles  triangle  the  bisector 
of  the  vertex  angle  is  the  perpendicular 
bisector  of  the  base. 

Fig.  77.  3.  If  two  angles  of  a  triangle  are  equal 


PARALLEL  LINES 


123 


the    sides    opposite  the  equal   angles    are    equal    and    the 
triangle  is  isosceles. 

106.  Parallel  Lines.  Straight  lines  in  the  same  plane 
that  will  not  meet,  however  far  they  are  extended,  are 
parallel  lines. 

1.  A  perpendicular  to  one  of  two  parallel  lines  (Fig.  78) 
is  perpendicular  to  the  other  also. 


Fig.  78. 


2.  If  two  parallel  lines  are  cut  by  a  third  straight  line, 
(Fig.  79): 


Fig.  79. 


(a)  The  alternate  interior  angles  are  equal 
(i.e.,  Z2=  Z7     and      Z4  =  Z5). 

(6)  The  alternate  exterior  angles  are  equal 
(i.e.,  Zl=  Z8     and      Z3=  Z6). 

(c)  The  corresponding  angles  are  equal 
(i.e.,  Z  1=  Z5,  etc.). 


124  GEOMETRY 

(d)  The  interior  angles  on  the  same  side  of  the  third 
line  are  supplementary 

(i.e.,  Z2+Z5  =  180°,  etc.). 

3.  If  two  lines  are  cut  by  a  third  line  making 

(a)  The  alternate  interior  angles  equal, 

(b)  The  alternate  exterior  angles  equal, 

(c)  The  corresponding  angles  equal,  or 

(d)  The  interior  angles  on  the  same  side  of  the  third  line 

supplementary, 
the  lines  are  parallel. 

Note.     No.  3  is  the  converse  of  No.  2. 

4.  Two  lines  perpendicular  to  the   same   straight   line 
(Fig.  80)  are  parallel. 


Fig.  80. 


107.  Quadrilaterals.  A  plane  figure  having  four  straight 
sides  is  a  quadrilateral. 

A  trapezium  is  a  quadrilateral  having  no  two  sides 
parallel. 


Trapezium  Trapezoid 

Fig.  81. 

A  trapezoid  s  a  quadrilateral  having  one  pair  of  parallel 
sides. 


QUADRILATERALS 


125 


Rhombus 

Fig.  82. 


A  quadrilateral  having  its  oppos'te  sides  parallel    s  a 
parallelogram. 

A  rectangle  is  a  parallelogram  having  four  right  angles. 

A  square  is  a  rectangle  having  four  equal  sides. 

A  rhombus  is  a  parallelogram 
having  four  equal  sides  and  four 
oblique  angles. 

108.  Parallelogram  Theorems. 

1.  The  opposite  sides  of  a  par- 
allelogram are  equal. 

2.  The  opposite  angles  of  a  parallelogram  are  equal. 

3.  Two  adjacent  angles  of  a  parallelogram  are  supple- 
mentary. 

4.  If  a  pair  of  opposite  sides  of  a  quadrilateral  are  equal 

and    parallel,    the    figure    is    a 
parallelogram. 

109.  Circles. 

1.  The  diameter  of  a  circle 
bisects  the  circle. 

2.  In  the  same  circle  or  in 
equal  circles,  radii  that  form 
equal  angles  at  the  center 
intercept  equal  arcs  on  the 
circumference,  and  conversely. 

FlG-  83-  That  is,  if  circle  I  is  equal 

to    circle    II    (Fig.  84),    and   if 


Zl=  Z2, 

then 

arc  l  =  arc  2 

or  if 

arc  1  =  arc  2, 

then 

Zl=  Z2. 

Fig.  84. 


126 


GEOMETRY 


3.  In  the  same  circle  or  in  equal  circles  equal  chords 
subtend  equal  arcs,  and  conversely. 

If  chord  A£  =  chord  CD  (Fig.  85),  then  arc  AB  =  arc  CD, 
and  conversely. 

4.  In  the  same  circle 
or  in  equal  circles  equal 
chords  are  equal  distances 
from  the  center  (Fig.  86), 
and  conversely,  if  AB  = 
CD,  then  OX=MN,  and 
conversely. 


Fig.  85. 


5.  A  line  fulfilling  any  two  of  the  following  conditions 
fulfills  the  other  two  (Fig.  87): 


Fig.  SO.  Fig.  87. 

(a)  Passes  through  the  center  of  the  circle. 

(6)  Bisects  the  chord. 

(c)  Is  perpendicular  to  the  chord. 

(d)  Bisects  the  arc. 


6.  A  tangent  to  a  circle  is  perpen- 
dicular to  the  radius  drawn  to  the 
point  of  tangency  (Fig.  88),  and  con- 
versely. 

7.  Arcs  included  between  parallel 
lines  are  equal. 


CIRCLE   THEOREMS  127 

8.  An  inscribed  angle  is  measured  by  one-half  the  inter- 
cepted arc  (Fig.  89). 


Fig.  89.  Fig.  90. 

9.  An  angle  formed  by  a  tangent  and  a  chord  Fig.  (90) 
is  measured  by  one-half  the  intercepted  arc. 

10.  An  angle  formed  by  two  secants  meeting  outside  a 
circle  (Fig.  91)  is  measured  by  one-half  the  difference  of  the 
intercepted  arcs. 


Fig.  91.  Fig.  92. 

11.  Two  tangents  drawn  to  a  circle  from  an  external 
point  (Fig.  92)  are  equal. 

110    Ratio  and  Proportion. 

1.  If  two  quantities  are  in  proportion  the  product  of 
the  extremes  equals  the  product  of  the  means: 

i.e.,  if  a  :b  ::  c  :  d, 

then  ad  =  be. 


128  GEOMETRY 

2.  A  line  parallel  to  one  side  of  a  triangle  divides  the 
other  two  sides  proportionally. 

3.  The  medians  of  a  triangle  meet  in  a  point  two-thirds 
of  the  distance  from  the  vertex  to  the  mid-point  of  the 
opposite  side. 

4.  The       bisector      of       the 
angle    of    a     triangle     (Fig.    93) 
divides    the    opposite    side    into 
segments     proportional     to    the 
Fig.  93.  adjacent  sides. 

111.  Similar  Triangles.  Triangles  having  their  sides 
proportional  and  angles  equal  are  similar. 

1.  Two  triangles  are  similar  if  two  angles  of  the  one 
are  equal  to  two  angles  of  the  other. 

2.  Two  triangles  are  similar  if  the  three  sides  of  the  one 
are  proportional  to  the  three  sides  of  the  other. 

3.  Triangles  having  their  corresponding  sides  parallel  or 
perpendicular  are  similar. 

112.  Right  Triangles. 

1.  Two  right  triangles  having  an  acute  angle  of  the  one 
equal  to  an  acute  angle  of  the  other  are  similar. 

2.  The  square  on  the  hypotenuse  of  a  right  triangle  is 
equal  to  the  sum  of  the  squares  of  the  other  two  sides. 

3.  The  perpendicular  from  the  vertex  of  the  right  angle 
of  a  right  triangle  to  the  hypotenuse  divides  the  triangle 
into  two  triangles  similar  to  each  other  and  similar  to  the 
original  triangle. 

4.  The  perpendicular  to  the  hypotenuse  of  a  right  triangle 
is  the  mean  proportional  between  the  segments  of  the 
hypotenuse. 

5.  Either   side   is   the   mean   proportional   between   the 


REVIEW   PROBLEMS 


129 


whole  hypotenuse  and  the  segment  of  the  hypotenuse  adja- 
cent to  that  side. 

EXERCISE    1 

1.  Find  the  other  angles  of  the  parallelogram,  Fig.  94. 

Ans.  5=126°. 

C  =  54°,  D=126°. 


Fig.  94. 

2.  Find  the  value  of  the  following  angles  in  the  regular 
inscribed  hexagon  Fig.  95: 


(a)  BOC. 

Ans.  60° 

(6)  BEC. 

30° 

(c)  AFE. 

120° 

(d)  ECD. 

30° 

F^ 

"vxE 

Fig.  95. 


3.  If  a  stick  10  ft.  long  casts  a  shadow  8  ft.,  how  high, 
is  a  tree  that  casts  a  shadow  55  ft.?  Ans.  68.7  ft. 


130 


GEOMETRY 


4.  In  the  right  triangles,  Fig.  96,  find  x. 

Ans.  x  =  4. 2896. 


10.724 


Fig.  96. 


5.  In  the  triangle,  Fig.  97,  find  the  hypotenuse. 

Ans.  20  ft. 


Fig.  97. 


6.  Find  the  segments  of  the  hypotenuse. 

Ans.  7|,  12i. 

7.  Find  the  perpendicular  to  the  hypotenuse.    Ans.  9|. 

8.  Find  the  area.  Ans.  96. 


CHAPTER  XII 
LOGARITHMS 

113.  Aids  to  Computation.  Various  means  have  been 
devised  for  simplifying  the  work  of  long  multiplication  and 
division,  such  as  necessarily  occurs  in  the  use  of  trigo- 
nometric functions  and  in  other  places.  Three  of  these 
means  are  in  common  use: 

I.  Calculating  Machines.  They  are  always  expensive 
and  not  easily  carried  about. 

II.  Slide  Rules.  Inexpensive  instruments  and  easily 
carried,  but  not  sufficiently  accurate  for  some  work. 

III.  Logarithms.  Tables  of  logarithms  are  very  nex- 
pensive  and  easily  carried  about  and  give  more  accurate 
results  than  the  slide  rule. 

Logarithms  were  first  put  into  practical  use  by  John 
Napier  in  1614,  later  they  were  improved  by  Henry  Briggs. 
Logarithms  are  used  extensively  in  computations  because 
they  reduce  multiplication,  division  and  finding  roots  to 
easy,  rapid  calculations.  Logarithms  are  also  necessary  in 
solving  certain  equations  and  are  used  in  some  formulas. 

114.  Common  Logarithms.  The  common  logarithm  of 
a  number  is  the  exponent  of  10  which  makes  the  power  of 
10  equal  to  that  number.  For  example,  102=100,  there- 
fore 2  is  the  logarithm  of  100. 

10  is  the  base  of  the  logarithm  in  this  definition;  other 
bases  are  used  in  higher  work.  The  student  must  keep  in 
mind  the  fundamental  conception  of  a  logarithm,  namely, 
that  "  a  logarithm  is  an  exponent." 

131 


132  LOGARITHMS 

115.  Finding    the    Logarithm    of    a  Number.     By  the 

definition  of  a  logarithm, 

102=100    therefore  log  100   =  2, 
103  =  1000  therefore  log  1000  =  3. 

Hence  the  logarithm  of  254  must  lie  between  2  and  3. 
That  is,  it  must  be  2+;  carried  to  five  places  it  is  2.40483. 
A  logarithm  is  composed  of  two  parts,  a  whole  number 
and  a  decimal.  The  decimal  part  of  the  logarithm  is  the 
mantissa  and  can  be  found  in  the  table  of  logarithms. 
The  mantissa  is  always  positive,  and  is  independent  of  the 
position  of  the  decimal  point  in  the  number.  That  is,  the 
mantissa  is  determined  by  the  figures  only.  For  example, 
the  mantissa  of  3472  is  the  same  as  the  mantissa  of  34.72. 

The  whole  number  part  of  a  logarithm  is  the  charac- 
teristic. The  characteristic  cannot  be  found  in  the  table, 
but  must  be  determined  by  rule.  The  characteristic  is 
sometimes  positive  and  sometimes  negative,  as  in  the  case 
of  decimals.  If  the  characteristic  is  —3  and  the  mantissa 
is  .44632,  the  whole  logarithm  cannot  be  written  —3.44632, 
as  this  would  indicate  that  the  mantissa  also  is  negative. 
The  mantissa  is  never  negative  and  so  the  following  nota- 
tion is  used,  3.44632,  which  means  that  the  characteristic 
alone  is  negative.*  The  student  should  learn  to  distin- 
guish between  characteristic  and  mantissa. 

116.  Finding  the  Characteristic.  The  characteristic  can 
be  determined  by  definition  thus: 

102=100      therefore  log  100     =2, 
103=1000    therefore  log  1000   =3, 
104  =  10000  therefore  log  10000  =  4. 
Hence  the  characteristic  of  the  logarithm  of  all  numbers 
*  Note.     The  following  notation  is  sometimes  used  for  negative 
characteristics:   7.44G32-10. 


CHARACTERISTIC 


133 


between  100  and  1000  is  2  and  of  all  numbers  between  1000 
and  10000  is  3,  etc.  If  the  number  of  figures  in  the  whole 
number  is  increased  or  decreased  by  1,  the  characteristic  of 
the  logarithm  of  the  number  is  increased  or  decreased  by  1. 
Hence: 

For  numbers  between  The  characteristic  is 

3 

2 

1 

0 

-1 

-2 

-3 

This  rule  is  too  cumbersome  for  efficient  use  so  the 
following  rule  will  be  used.  Observe  the  following  table  of 
numbers  and  their  logarithms: 

Number.  Logarithm. 


1000 

and  10000 

100 

and  1000 

10 

and  100 

1 

and  10 

.1 

and  1 

.01 

and   .  1 

.001 

and  .01 

0.00254  3.40483 

0.0254  2.40483 

0.254  1.40483 

2.54  0.40483 

2  5.4  1.40483 

25  4  .  2.40483 

254  0.  3.40483 

2540  0.  4.40483 

Observe  that  the  characteristics  in  the  above  table 
could  be  found  by  counting  from  the  units  figure  (the 
figure  in  the  column  between  the  lines)  to  the  2. 

Rule.  To  find  the  characteristic  count  from  {not  includ- 
ing) the  units  figure  to  the  first  significant  figure. 


134  LOGARITHMS 

Observe  that: 

1.  If  the  count  is  to  the  left,  the  characteristic  is  positive, 
and  if  to  the  right,  the  characteristic  is  negative. 

2.  A  significant  figure  means  a  figure  other  than  zero,  and 
the  first  significant  figure  means  the  first  figure  (other  than 
zero)  on  the  left  of  a  number. 

Example  1.     What  is  the  characteristic  of  7934?     Count 

3  2  1 

from  the  4  to  the  7,  thus  7934.     The  characteristic  is  3. 
Example   2.     What  is  the   characteristic   of  0.000853? 

Count   from   the  first   0   to  the   8,    thus    0.000853.      The 
characteristic  is  —4. 

EXERCISE    1 

What  is  the  characteristic  in  the  following  numbers? 

1.  25  6.  0.0623 

2.  354  7.  0.00043 

3.  3540  8.  0.0004 

4.  75.84  9.  0.421 

5.  66319  10.  6.3 

117.  Finding  the  Mantissa.  The  mantissa  must  be 
found  in  the  table  of  logarithms.  The  first  two  figures  of 
a  number  will  be  found  in  the  column  marked  N  at  the 
left  of  the  page  and  the  third  figure  will  be  found  at  the 
top  of  the  page.  The  mantissa  will  be  found  in  the  row 
with  the  first  two  figures  and  in  the  column  of  the  third 
figure. 

Example  1.  Find  the  mant  ssa  o  567.  Look  for 
56  in  the  column  at  the  left  of  the  page  and  find  the 
mantissa  (.75358)  in  that  row  and  in  the  column  headed  7. 
The  complete  logarithm  is  2.75358. 

In  finding  a  logarithm  the  following  steps  should  be 
followed: 


INTERPOLATION  135 

1.  Write  the  number  thus,  log  567  = 

2.  Determine  the  characteristic,  log  567  =  2. 

3.  Find  the  mantissa,  log  567  =  2.75358. 

For  efficiency  in  finding  the  mantissa,  fix  the  entire 
mantissa  in  mind  before  writing  any  of  it. 

EXERCISE   2 

Find  the  logarithms  of  the  following: 

1.  567.  7.  9. 

2.  983.  8.  0.234. 

3.  75.4.  9.  0.0056. 

4.  6330.  10.  7.03. 

5.  565.  11.  40.2. 

6.  230.  12.  768. 

118.  Interpolation.  The  logarithm  of  4683  cannot  be 
found  in  an  ordinary  logarithm  table,  but  the  logarithms 
46S0  and  4690  can  be  found.  The  logarithm  of  4683  must 
be  determined  by  interpolation  between  these  two  logarithms. 
The  work  can  be  arranged  as  follows: 

log  4680  =  3. 67025 
log  4690  =  3. 67 117 


92  (Tabular  difference) 

Take  .3  of  this  difference  and  add  to  the  logarithm  of 
4680,  thus  .3  X  92  =  28,  that  is,  28  in  the  last  two  places. 

log  4680  =  3. 67025 

+28 


3. 67053  =  log  4683 

In  general,  if  the  fourth  figure  of  a  number  is  different 
than  3,  as  7,  take  .7  of  the  tabular  difference  and  add  to  the 


136  LOGARITHMS 

logarithm  of  the  smaller  number.  In  interpolation  do  not 
carry  the  results  beyond  the  fifth  place. 

The  above  method  is  too  elaborate  for  rapid  efficient 
use  of  logarithms,  and  for  efficiency,  interpolation  must  be 
done  mentally.  The  following  steps  and  suggestions  should 
prove  helpful: 

Example  1.     Find  the  logarithm  of  753.4. 

I.  Write  the  characteristic. 

II.  Look  for  the  mantissas  of  753  and  754;  they  will 
be  found  side  by  side,  thus:   87679  and  87737. 

III.  Subtract  mentally,  getting  the  difference  of  58  (in 
the  last  two  places). 

IV.  .4X58=23.2  or  23  in  the  last  two  places. 
V.  87679+23  =  87702. 

VI.  Write  the  mantissa  87702  after  the  characteristic. 

Note.  Tables  in  this  book  will  give  results  correct  to  three  or  four 
figures.  Students  desiring  more  accurate  results  can  obtain  more  com- 
plete tables  in  a  small  compact  book  at  a  moderate  price. 

EXERCISE   3 

Find  the  logarithms  of  the  following: 

1.  6346.  Ans.  3.80250. 

2.  8437.  3.92619. 

3.  245.2.  2.38952. 

4.  3.657.  0.56312. 

5.  5637.  1.75105. 

119.  Multiplication  by  Logarithms.  Recall  the  funda- 
mental part  of  the  definition  of  a  logarithm,  namely,  that 
"  A  logarithm  is  an  exponent,"  and  recall  the  law  of  expo- 
nents for  multiplication  whereby  x3Xx4  =  x~.  It  will  be 
seen  that  the  sum  of  the  logarithms  of  two  numbers  is  the 


ANTILOGARITHMS  137 

logarithm   of  their  product.     Therefore  the  following  rule 
can  be  stated: 

Rule.  To  multiply  two  numbers  add  their  logarithms  and 
find  the  number  corresponding  to  the  sum. 

120.  Antilogarithms.  The  number  which  corresponds  to  a 
given  logarithm  is  called  its  antilogarithm. 

Example  1.  Find  the  antilogarithm  of  2.94576.  Look 
for  the  mantissa  (94596)  in  the  main  part  of  the  table. 
The  first  two  figures  of  the  number  (88)  will  be  found  in 
the  column  marked  N  at  the  left  side  of  the  page,  in  the 
row  in  which  94596  is  found.  The  third  figure  (3)  is  found 
at  the  top  of  the  column  in  which  94596  is  found.  There- 
fore the  figures  corresponding  to  the  mantissa  94596  are 
883.  The  position  of  the  decimal  point  is  determined  by 
the  characteristic.  The  method  is  the  opposite  of  the 
method  of  finding  the  characteristic.  For  example,  in  the 
above  problem  the  characteristic  is  2  and  the  figures  are 
883.     Therefore,  count  2  from  the  8  thus: 

12 

883 
The  next  figure  is  units  thus: 

'a 
12  3 

883 

Rule.  Count,  from  the  first  figure  of  the  number  obtained, 
as  many  places  as  indicated  by  the  characteristic,  the  next 
figure  will  be  units.  If  the  characteristic  is  positive,  count  to 
the  right,  and  if  the  characteristic  is  negative,  count  to  the 
left.     Zeros  will  have  to  be  supplied  in  this  case. 

If  the  mantissa  cannot  be  found  exactly  in  the  tables, 
interpolate  between  the  two  numbers  corresponding  to  the 
two  nearest  logarithms. 


138  LOGARITHMS 

Example.     Find  the  antilog  of  3 .  56505. 

56467  =  log  367 
56585  =  log  368 
(Tabular  difference)      118 

56467  =  log  367 

56505  =  log  required  number 

Difference  38 

Diff.      =J38  = 
Tab.diff.     118         + 

Therefore    56506  =  log  3673T\%  =  log  3673  + 
The  figures  of  the    antilogarithm   are  3673,  the  charac- 
teristic is  3,  therefore  count  as  follows: 


3321 
0.003673.  Ans. 

In  interpolation  results  are  not  carried  more  than  one 
place  beyond  the  figures  given  in  the  table. 

EXERCISE   4 

Find  the  antilogarithms  of  the  following  logarithms: 

1.  2.58286.  Ans.  382.7. 

2.  4.59106.  39000. 

3.  0.76384.  5.8055. 

4.  2.84365.  .069767. 

5.  1.08458.  12.15. 

6.  I. 23615.  .17225. 

7.  0.29336.  1.965. 

8.  5.25527.  180000. 


MULTIPLICATION 


139 


121.  Multiplication.     The    work    of    multiplication 
logarithms  should  be  arranged  as  follows: 
Multiply  3675  by  78 .  56. 

log   3675  =  3.56526 
log  78.56=  1.89520 


by 


Ans. 


5.46046  (Adding) 
288710.  (Antilog) 


EXERCISE   5 


Perform    the    following    multiplications    by    means    of 


logarithms : 

1.  243X375. 

2.  3984X5.6 

3.  .024X7685. 

4.  . 654 X. 4379. 

5.  76X5. 

6.  24765X3.49. 

7.  .0024X76587. 

8.  .653X378X55.15. 

9.  273X68.49X23.69. 

10.  .  002  X.  0347X56. 

11.  1 27000  X.  5634. 

12.  653.14X5.7823. 


Ans.  91126. 
22310. 
184.44. 
.28639. 
380. 
86432. 
183.81. 
13613. 
442960. 
.0038865. 
71552. 
3776.7. 


122.  Division.     Since 

sy*0      ■      /yd  —   /wi 
•*/  •     •*•       — ~~  •*/      • 

and  since  a  logarithm  is  an  exponent,  the  logarithm  of  the 
quotient  of  two  numbers  is  the  difference  of  the  logarithms 
of  the  numbers.  The  logarithm  of  the  divisor  is  to  be 
subtracted  from  the  logarithm  of  the  dividend.  The  anti- 
logarithm  of  the  difference  is  the  quotient. 


140 


LOGARITHMS 


123.  Cologarithms.  The  above  method  of  division  is 
too  inefficient  when  several  numbers  are  to  be  multiplied 
and  divided.     For  example: 

254X781 
53X125 


Add: 


Add: 


log  254  =  2.40483 
log  781  =  2.89265 


5.29748 


log    53=1.72428 
log  125  =  2.09691 


3.82119 


Subtract: 


Therefore : 


5.29748 
3.82119 

1.47629  =  log  29.943 
254X781 


53X125 


29.943. 


9^4- V7R1 
But      „*/°      can    be    written    254X781  X^Xy^, 
53  X  l<iO 

and,  if  the  logarithms  of  ^,  and  y^g  can  be  found  readily, 

this  problem  reduces  to  multiplication,  and  hence  only  one 

addition  is  necessary:  ^  =  1  -s-53. 

log5-V  =  logl-log53 


log 


0       -log  53     (log  1  =  0) 


log5V=-l°g53 
The  work  can  be  arranged  as  follows: 

Subtract : 

0 
log  53=  1.72428 


COLOGARITHMS  141 

Borrow  1  from  0  leaving  —  1 ,  thus : 

T.999910 
1.72428 


2.27572 


The  logarithm   of  —  -, —  is  the  cologarithm  of  the 

a  number 

number.     The  above  problem  now  reduces  to: 

Add:  log  254  =  2.40483 

log  781  =  2.89265 
colog  53  =  2.27572 
colog  125  =  3.90309 


1.47629  =  log  29.943.  Ans. 

For  efficiency  the  mantissa  of  the  cologarithm  must  be 
obtained  mentally,  from  the  tables.  The  work  can  be  done 
as  follows:  Find  the  mantissa  of  the  logarithm,  for  example 
34624,  subtract  each  of  the  first  four  figures  from  9  and 
the  last  one  from  10,  fix  these  numbers  in  mind  and  write 
all  down  at  once  thus,  65376.  For  the  characteristic 
remember  the  T  in  the  minuend  and  the  laws  for  algebraic 
subtraction.  As  the  student  becomes  more  familiar  with 
the  work  the  characteristic  of  the  cologarithm  can  be  found 
as  readily  as  the  characteristic  of  the  logarithm. 

Ruls.  To  divide  numbers  by  means  of  logarithms  add 
the  logarithms  of  all  the  dividends  and  the  cologarithms  of  all 
the  divisors  and  find  the  antilogarithm  of  the  sum. 

The  work  should  be  arranged  as  follows; 

246X67.85 
Example.  23. 5  X.  04732" 


142  LOGARITHMS 

Solution. 


log 

246       = 

: 2. 39094 

log 

67.85   = 

=1.83155 

colog 

23 . 5     = 

=2.62893 

colog 

.04732= 

=1.32496 

4.17638 

Ans. 

15010. 

EXERCISE   6 

wing 

by  logarithms: 

Ans.  23 

.743. 

234X451 
635X7  ' 

2  49^82X6^731  2  9864 

5.347X21 

3  6542X31^67  %> .054. 

0.427X167' 

4.  i0256*68-715.  .000051085. 


762X45.19 

3.1416X258.9 

275X.02     " 

6543X289 
21. 32X1. 318' 


147.88. 
67293. 


35X24.32  0.59525. 

65X22 

8  J65X694J 223  Q4 

23.4X4.675X46.75 

124.  Powers  of  Numbers :  | 

52  =  5x5  =  25 
Therefore, 

log  52  =  log  5+log  5  =  2  log  5 


POWERS   AND   ROOTS  143 

Similarly, 

log  53  =  log  5+log  5+log  5  =  3  log  5 

From  which  the  following  rule  can  be  stated: 
Rule.     To  raise  a  number  to  a  power  multiply  the  logarithm 
of  the  number  by  the  exponent  of  the  number  and  find  the  anti- 
logarithm. 

Example.     Find  373. 

log  37=  1.56820 
3 


4.70460 
373  =  50652 

EXERCISE  7 

Find  by  means  of  logarithms: 

1.  65.  Ans.  7776. 

2.  .374.  .018741. 

3.  5.373.  154.85. 

4.  1.2573.  1.9862. 

5.  .12543.  0.001972. 

125.  Roots  of  Numbers. 

25  =  52 

log  25  =  2  log  5 

V25  =  5 

log  V25  =  log  5 

2  log  V25  =  2  1og5 


144  LOGARITHMS 

But 


Therefore 


Similarly, 


log  25  =  2  log  5 
2  log  \/25  =  log25 
log  V25  =  |  log  25 
log  \/Qi=  I  log  64 


From  the  above  the  following  rule  can  be  stated: 

Rule.  To  find  the  square  root  of  a  number  divide  the 
logarithm  of  the  number  by  2  and  find  the  antilogarithm. 

To  find  the  cube  root  of  a  number  divide  the  logarithm  by  3 
and  find  the  antilogarithm. 

To  find  any  root  of  a  number  divide  the  logarithm  of  the 
number  by  the  index  of  the  root  and  find  the  antilogarithm. 

Note.  The  index  of  a  root  is  the  small  number  placed  above  and 
to  the  left  of  the  radical  sign,  and  indicates  which  root  is  to  be  taken. 
For  example,  in  \^75,  3  is  the  index  of  the  root. 

Example  1.     Find  \/lb. 

log  75=1.87506 
^3=    .62502 
_4.2172=-^75 
Example  2.     Find  \Z.05. 

log  .05  =  2.69897 

2.69897  cannot  be  divided  by  3  as  in  the  previous 
problem,  since  3  will  not  go  in  2  evenly,  and  a  negative 
remainder  cannot  be  carried  over  into  the  positive  mantissa. 
But 

2.69897  can  be  written 

-2+    .69897, 


UNKNOWN    EXPONENTS 


145 


or 


Then 


or 
hence 


-3  +  1.69897. 


-3  +  1.69897 


-1+    .56632 
1 . 56632 
^.05  =  .3684 


EXERCISE   8 


Find  the  following: 

1.  ^50. 

2.  ^3775. 

3.  V73. 

4.  V8700. 

3 


6I(  34.28X57.5  V 
\\48. 08X5. 962/ 


Ans.     3.684. 
5.1614. 
.54773. 
93.274. 

3.6161. 


126.  Equations  Containing  an  Unknown  Exponent. 
Axiom.     If  two  quantities  are  equal,  their  logarithms  are 
equal.     Therefore  if 

3X  =  81 

log3z  =  log81 

x  log  3  =  log  81 

xX.  47712  =1.90849 

1.90849 


x  = 


.17712 


z  =  4 


146 


I 

LOGARITHMS 

EXERCISE  9 

Solve  for  x: 

1.  5*     =625. 

2.  3J     =243. 

3.  2J     =19. 

4.  4J  +  1  =  256. 

Ans.  4. 
5. 

4.2479 
3. 

exercise  10 
Miscellaneous  Problems 

1.  Given  F  =  f7ir3  as  the  formula  for  the  volume  of  a 
sphere,  where  r  is  the  radius.  Find  the  volume  of  a  sphere 
having  a  radius  of  13.34  in.  Ans.  9944.3. 

2.  Find  the  diameter  of  a  steel  ball  weighing  16  lbs.  if 
steel  weighs  .283  lb.  per  cubic  inch.  Ans.  4.762. 

3.  Evalute  the  formula 

C=—         -'  (Capacity  of  aerial  wire) 

when  s=24     and     r  =  \.       Ans.   .0019574. 

4.  The  formula  for  compound  interest,  compounded 
annually,  is  ^4.=a(l  +  r)",  where  A  =  the  amount  at  the 
end  of  n  years,  a  is  the  original  investment,  r  is  the 
rate,  and  n  the  number  of  years.  Find  the  amount  of 
$150.00  at  the  end  of  20  years,  interest  compounded  annually 
at  4  per  cent.  Ans.  $328.62. 


REVIEW   PROBLEMS  147 

5.  How  many  years  will  it  take  $100.00  to  amount  to 
$240.66?     Interest  compounded  annually  at  5  per  cent. 

Ans.  18  years. 

6.  Evaluate  the  formula, 

L=  .0017XZXlog  p    (Inductance   in   a   trans- 
mission line) 
when  I  =  25,  d  =  18,  and  R  =  .25.  Ans.  .07894. 

7.  Evaluate  the  formula, 

2  41 3  Kl 

C  =  — ^     (Capacity  of  a  submarine  cable) 

lonog^ 

whenX  =  6,  Z  =  1000,  D=2.5,  d  =  1.5.  Ans.  0.00666. 

8.  Evaluate  the  formula  for  the  capacity  of  a  submarine 
cable  when  C  =  .  25,  I  =  35,000,  D  =  3,  d  =  1 .  75. 

Ans.  6.929. 


CHAPTER  XIII 

RIGHT  TRIANGULATION 

127.  Constant    Ratios    in  Right  Triangles.     The  right 
triangles  shown  in  Fig.  98  are  similar  since  they  have  an 


Fig.  98. 


acute  angle  of  the  one  equal  to  an  acute  angle  of  the  other. 


CL\  _  0,2  _  as 
C\        Co        C3 


(why?) 


Therefore  all  right  triangles  having  one  acute  angle  equal 
to  23°  are  similar,  and  therefore  the  ratio  of  the  side  opposite 
23°  to  the  hypotenuse  in  each  triangle  is  constant  (always 
the  same).  By  actual  measurement  of  the  sides  this  ratio 
reduced  to  a  decimal  correct  to  .00001  can  be  calculated  to 
be  0.39073. 

148 


CONSTANT    RATIOS 


149 


In  the  same  manner: 
bi  _  62 
ci     c2 


C3 


0.92050 


£l  =  ^  =  ^  =  0. 42447 
01      02     03 


2 .  3558.5 


b\  _  bo  _  63 

d\       0,2       0,3 


***     1.0884 

Oi      62      03 

*=.*=*  =  2.5fiB3 


«i     do     as 
Example  1.     In  the  right  triangle  of  Fig.  99, 


™  =  0.39073 
bO 

a  =  23. 4438 


^r=   0.9205 
bO 


60. 


23 


90 


b 
Fig.  99. 


6  =  55.230 
Example  2.     In  the  right  triangle  of  Fig.  100, 


a 
42 

a=? 


=  0.42477 


ts=  1.0864 
42 

/>  =? 


Observe  that:    a  z's  the  side  opposite  23°,  b  is  the  side 
adjacent  to  23°,  and  c  is  he  hypo  enuse  of  he  right  triangle. 


150 


RIGHT  TRIANGULATION 


For  ihe  purpose  of  abulat'ng  these  constant  ratios  it  is  con- 
venient to  refer  to  them  by  names      Referring  to  Fig.  101, 


Fig.  101. 


a  . 


is  called  the  s*ne  of  23°,  written  sin  23c 


b  . 


is  called  the  cosine  of  23°,  written  cos  23 c 


a  . 


T  is  called  the  tangent  of  23°,  written  tan  23 c 
o 


-  is  called  the  cotangent  of  23°,  written  cot  23°. 
a 


is  called  the  secant  of  23°,  written  sec  23°. 


c  . 


is  called  the  cosecant  of  23°,  written  esc  23c 


sin 


cos 


tan  cot  sec  esc 


23°    0.39073 


0.92050 


0.42447 


2.3559  I  1.0864 


2.5593 


These  ratios  can  be  calculated  for  any  angle  and  are 
called  functions  of  the  angle.  In  general,  for  any  value  of 
angle  A, 


FUNCTIONS   DEFINED  151 

.     A     a     opposite  side 

sin  A  =  -  =  -  — 

c      hypotenuse 

.      b    adjacent  side 

COS  A  =  —  = 

c      hypotenuse 

.  _  a  _  opposite  side 
b    adjacent  side 

,  .      6     adjacent  side 

COt   A  =  -  = ; — 

a     opposite  side 

A      c       hypotenuse 

sec  A  =  -  =  ~^- — 

6    adjacent  side 

.  _  c  _  hypotenuse 

CSC  rx ■ 


opposite  side 

By  using  a  table  of  these  ratios  for  all  angles,  if  one  side 
and  one  acute  angle  of  a  right  triangle  are  known  all  the 
remaining  parts  can  be  calculated.  Table  IV  contains 
these  ratios  for  all  angles. 

128.  Use  of  the  Tables. 

Example  1.  Find  the  sine  of  37°  20'.  Locate  the 
sine  page  (Table  IV).  Find  37°  in  the  column  at  the  left. 
The  sine  of  37°  20'  is  found  in  the  row  marked  37°  and  in 
the  column  marked  20'  at  the  top. 

Example  2.  Find  the  cosine  of  52°  50'.  Find  the 
cosine  page  (Table  IV).  (Cosine  is  written  at  the  bottom 
of  the  page.)  Find  the  degrees  in  the  column  at  the  right 
and  the  minutes  at  the  bottom  of  the  page. 

Observe  that:  To  find  the  sin  or  tan,  locate  the  angle 
in  degrees  at  the  left  of  the  page  and  the  minutes  in  the 
column  at  the  top.  To  find  the  cos  or  cot,  locate  the 
angle  in  degrees  at  the  right  of  the  page  and  the  minutes 
in  the  column  at  the  bottom. 


152  RIGHT  TRIANGULATION 

Example  3.     Find  a  and  6  in  Fig.  102. 


65/S 

a 

—  =  sin  37°. 
b5 

/ri° 

£  =  .60181. 
b5 

b 

Fig.  102. 

a  =  39.1176. 

To  find  6: 

£=  =  cos  37°. 
65 

£=.79863. 
b5 

6  =  51.9109 

To  find  ZB: 

ZA+ZB  =  90°. 

37< 

'4-/5  =  90°. 
ZJ3=53°. 

Check. 

c2=a2+62 

652  = 

=  39.11762+51.91092 

4225  = 

=  1530.2+2694.8+ 

4225  = 

=  4225  + 

Note.     Tables  correct  to  five  figures  will  give  results  correct  to 
five  figures. 


USE   OF   THE   FUNCTIONS  153 

Example  4.     Find  6,  c,  and  Z  A  in  the  triangle  of  Fig.  103. 
To  find  ZA: 

ZA+Z£=90°. 
ZA+Z59°  =  90°. 
ZA=31°. 
To  find  b: 

rwt31°- 

A=  1.6643. 
6  =133.144. 

?°  =  sin31°. 
c 

-=.51504. 

c 

80  =  . 51504c 
c=  155.328. 

Observe:    To  find  b,  T  =  tan  A  might  have  been  used, 

which  would  give 

SO 

^  =  .60090. 
o 

,         80 
.60090* 


To  find  c: 


6=133.1  + 


154 


RIGHT  TRIANGULATION 


By  selecting  the  ratio  which  has  the  unknown  side  for 
numerator,  the  result  is  obtained  by  multiplying  by  a 
decimal  rather  than  by  dividing  by  a  decimal. 

EXERCISE    1 

Find  x  in  the  triangles  of  Fig.  104: 


(a) 
(a)  Ans.  58.084. 


(&) 
(b)  Ans.  90.308. 


42 
(c)  Ans.  31.649. 


340 

(d) 

(d)  Ans.  349.18. 


(e)  Ans.  236.84. 
Fig.  104. 

2.  The  diagonal  of  a  rectangle  is  48"  and  the  angle 
between  it  and  the  base  is  28°.  Find  the  dimensions  of  the 
rectangle.  Ans.  22.535;  42.361. 


TWO   SIDES   GIVEN 


155 


3.  The  altitude  of  an  isosceles  triangle  is  15"  and  the 
base  angles  are  37°  5'.     Find  the  sides  of  the  triangle. 

Ans.  24.876;  24.876;  39.690. 


a/ 

<N 

1 

\b 

/±2° 

32 

40*\ 

Fig. 

105. 

Find  a  and  b  in  the  trapezoid  of  Fig.  105. 

Ans.  a  =  35.868;  6  =  44.465. 

5.  The  altitude  of  an  isosceles  triangle  is  24"  and  the 
angle  at  the  vertex  is  97°  28'.     Find  the  sides  of  the  triangle. 

Ans.  36.386;  36.386;  54.701. 

6.  The  angle  between  the  two  sides  of  a  parallelogram 
is  43°  15'  and  one  of  the  sides  is  3.5"  long.  Find  the  distance 
between  the  other  two  parallel  sides.  Ans.  2.3981. 


129.  Solution   of  a  Right  Triangle,  Two  Sides  Given. 

If  two  sides  of  a  right  triangle  are  known,  all  the  remaining 
parts  can  be  calculated.  The  third  side  can  be  found  by 
the  formula  c?  =  a?-\-b2  and  the  angles  can  be  found  from 
Table  IV. 

Example  1.     Find  b,  Z  A  and  Z  B  in  the  right  triangle 
of  Fie.  106. 


203.37 


To  find  ZA: 

203.37      .     , 
500    "Sm^- 

j$^^ 

.40674  =  sin  A. 

A 

b 

.40674  =  sin  24°. 

Fig.  106. 

156 


RIGHT  TRIANGULATION 


To  find  6: 


ZA  =  24°. 

ZB  =  90°  -24°  =  66°. 


500  =  COS24°- 


500  = -91355' 
6  =  456.775. 
Note,     b  could  have  been  found  by  using  the  formula  c2  =  a2+b2. 
Example  2.     Find  Z  A  in  the  right  triangle  of  Fig.  107. 

!-  =  tan  A. 


48.23 


48.23  , 

—  =  tan  A. 


25 
1.9292  =  tan  A. 
1.9292  =  tan  62°  36'. 
ZA=62°  36'. 
Example  3.     Find  Z  B  in  the  triangle  of  Fig.  108. 

40  R 

-  =  cos  B 


70.13 
.57036  =  cos  B. 
Z£  =  55°  13'. 


Fig.  108. 


Note.     When  the  decimal  does  not  exactly  correspond  to  one 
the  table,  the  required  angle  can  be  expressed  in  degrees,  minutes  ; 
seconds.     For  the  present,   angles  will  be  expressed  to   the  no 
minute. 


PROBLEMS 


157 


EXERCISE   2 

1.  Find  ZA,  Fig.  109.  2.  Find  Zx,  Fig.  110. 


Fig.  109. 
Ans.  48°  11'. 


24.8 

Fig.  110. 

Ans.  71°  42'. 


3.  Find  x,  Fig.  111. 


32 

Fig.  111. 
Ans.  52.802. 


4.  Find  x,  Fig.  112. 


Fig.  112. 
Ans.  41°  20'. 


5.  The  base  of  an  isosceles  triangle  is  21"  and  the  equal 
sides  are  each  16".     Find  the  base  angles.       Ans.  48°  59'. 

6.  The  base  of  an  isosceles  triangle  is  48"  and  the  alti- 
tude is  32".     Find  the  vertex  angle.  Ans.  73°  44'. 

7.  The  dimensions  of  a  rectangle  are  78"  and  64". 
Find  the  angle  included  between  the  diagonal  and  the 
longer  side.  Ans.  39°  22'. 

8.  One  side  of  a  parallelogram  is  2.25"  and  the  distance 
between  the  other  two  parallel  sides  is  1.75".  Find  the 
angles  of  the  parallelogram.  Ans.    51°  3'. 


158  RIGHT  TRIANGULATION 

9.  The  diagonals  of  a  rhombus  are  58"  and  76".  Find 
the  angles  of  the  rhombus.  Ans.  74°  42';  105°  18'. 

10.  The  non-parallel  sides  of  a  trapezoid  are  48"  and 
54"  and  the  altitude  is  36".  Find  the  angles  of  the  trape- 
zoid. Ans.  48°  35',  131°  25';  41°  49',  138°  11'. 

130.  Interpolation.  Most  tables  of  the  functions  of 
angles  give  the  values  for  degrees  and  minutes  only,  or  to 
every  ten  minutes.  When  a  function  of  an  angle  involving 
seconds  is  to  be  used  it  must  be  determined  from  the 
functions  of  the  two  nearest  angles  given. 

Example  1.     Find  sin  27°  27'. 
From  the  tables, 

sin  27°  30'  =.46 175 
sin  27°  20' =.459 17 


Difference  for  10'  =  .00258   (tabular  difference) 

Difference  for  7'  is  ^  of  .00258  =  .00181. 

Adding  this  difference  to  sin  27°  20' =  .459 17 


gives  .46098  =  sin  27°  27' 


Example  2.     Find  tan  44°  16'. 

tan  44°  20' =97700 
tan  44°  10' =97 133 


567  (Tabular  difference) 
T6o  of  567  =  340  (in  the  last  3  places) 

tan  44°  10' =97 133 
+  340 


=  97473  =  tan  44°  16' 


INTERPOLATION  159 


Example  3.     Find  cos  17°  22'. 

cos  17°  20' =.95459 
cos  17°  30' =.95372 


87    (Tabular  difference) 
i%  of  87=17 
cos  17°  20' =95459 
-17 

.95442  =  cos  17°  22' 

Observe  that:  1.  The  cosi?ie  and  the  cotangent  of  an 
angle  vary  inversely  as  the  angle. 

2.  In  interpolation  for  the  cosine  (or  cotangent)  the  frac- 
tional part  of  the  tabular  difference  must  be  subtracted  from 
the  cosine  (or  cotangent)  of  the  smaller  angle. 

3.  When  a  five-place  table  is  used  in  interpolation  the 
result  is  never  carried  beyond  the  fifth  place. 

For  efficiency  in  interpolation  the  tabular  difference  must 
be  found  mentally.  The  fractional  part  of  the  tabular 
di [Terence  can  generally  be  determined  mentally.  This 
difference  should  be  added  to  (or  subtracted  from)  the 
function  of  the  smaller  angle  mentally. 

EXERCISE  3 

Find 

1.  sin  27°  15'.  Ans. .  45787. 

2.  sin  65°  29'.  .90984. 

3.  tan  32°  41'.  .64158. 

4.  cos  43°  12'.  .72897. 

5.  tan  28°  32'.  .54371. 

6.  cot     6°  32'.  8.73262. 

7.  tan  85°  17'.  12.1201. 

8.  cos  64°  18'.  .43365. 


160  RIGHT  TRIANGULATION 

131.  Finding  the  Angle  that  Corresponds  to  a  Given 
Function. 

Example  1.     Find  Z  x  if  tan  x  =  .63707. 

Find  63707  on  the  tangent  page  of  Table  IV.  It  will 
be  found  to  lie  in  the  32°  row  and  30'  column.     Therefore 

Z.r  =  32°  30' 

Example  2.     If  sin  a;  =  .23137,  find  Zx. 

Look  for  .23137  on  the  sin  page  of  Table  IV.     It  cannot 

be  found,  but 

.23062  =  sin  13°  20' 

.23345  =  sin  13°  30' 


Tabular  diff.       =       283 

Therefore  x  lies  between  13°  20'  and  13°  30': 

23062  =  sin  13°  20' 

23137  =  sin  x 


Difference  =         75 

Therefore  x=  13°  20'+^  of  10'  =  13°  23'. 

In  practice  the  work  should  be  done  mentally. 

EXERCISE  4 
Find  x  if: 

1.  sin  x=    .63910.  Ans.  39°  44'. 

2.  sin  x=    .76912.  50°  17'. 

3.  cos  x=    .80653.  36°  15'. 

4.  cos  x=    .43347.  64°  19'. 

5.  cot  z  =  6.0776.  9°  20'. 

6.  tan  re  =  5. 3977.  79°  30'. 

132.  Logarithms  of  the  Functions  of  Angles.     Table  III 
gives  directly  the  logarithms  of  the  trigonometric  functions 


ELEVATION    AND    DEPRESSION  161 

of  angles.  The  table  is  used  the  same  as  Table  IV.  The 
characteristic  of  the  logarithm  will  be  found  in  the  table. 
Table  III  makes  it  possible  to  multiply  and  divide  by 
trigonometric  functions  by  means  of  logarithms  without 
the  use  of  two  tables.  It  is  recommended  that  logarithms 
be  used  in  the  remainders  of  the  problems  wherever 
possible. 

133.  Angle    of    Elevation    and    Depression.     When    an 
object  is  sighted  above 
the   horizontal  plane, 
the     angle     which     a 

line  from  the  eye   to  ^'" 

the  object  makes  with  ''mevation 

the    horizontal    plane 

/\         "--   Depression 

is  the  angle  of  elevation, 


(Fig.  113).     When  an 

object  is  sighted  below 

the  horizontal    plane,  Fig.  113. 

the  angle  which  a  line 

from  the  eye  to   the    object    makes    with    the    horizontal 

plane  is  the  angle  of  depression. 

EXERCISE  5 

1.  Find  the  angle  of  elevation  of  the  sun  when  a  pole 
20'  high  casts  a  shadow  14'  9"  long.       Ans.  53°  35'  30". 

2.  A  building  145'  high  forms  an  angle  of  elevation  of 
18°  3'  from  a  point  on  level  ground.  How  far  is  the 
observer  from  the  building?  Ans.  444.95. 

3.  The  distance  from  the  foot  to  the  top  of  the  hill  is 
220  yds.  Find  the  height  of  the  hill  if  it  is  of  uniform 
grade  of  7°  2'.  Ans.  26.94. 


162 


RIGHT  TRIANGULATION 


4.  The  pitch  of  a  roof  is  40°,  the  width  of  the  building 
is  28  ft.  What  length  rafter  must  be  used  if  they  project 
18  in.  beyond  the  sides?  Ans.  19.77. 

5.  Find  the  height  of  a  tree 
if  it  makes  an  angle  of  25°  35' 
at  a  point  25  ft.  from  the  base 
of  the  tree.  Ans.  11.96. 

6.  Find  the  base  angle  and 
the  altitude  of  a  cone  if  the 
slant  height  is  8  in.  and  the 
radius  of  the  base  is  6  in. 

Ans.  41°  24'  34";  5.2915. 

7.  ABCD-O,  Fig.  114,  is  a 
square  pyramid  having  an  edge 

of  15  in.  and  one  side  of  the  base  10". 


(a)  Find  the  angle  OCM. 

Ans.   70°  32' 

(b)  Find  the  slant  height  OM. 

14.142. 

(c)   Find  the  angle  OMP. 

90°. 

(d)  Find  the  angle  OCP. 

61°  52' 

26 

(e)   Find  the  altitude  OP. 

13.299. 

Find  BC,  Fig.  115. 

Fig.  115. 


PROBLEMS 


163 


Let 


and 
and 


y  =  AC 
z  =  CD 
x  =  CB 

y+z  =  215 

V-  =  cot  32°  10' 

x 

2/=1.5900x 
~  =  cot41°22' 


2500 
Fig.  116. 


x 

2=1.1356* 
1. 5900s  4- 1.1356a; =215 

2. 7256a:  =  215 
z  =  7S.88 

9.  Two  persons  at  A  and 
B,  Fig.  116,  observe  an  air- 
plane directly  over  the  line 
between  them.  If  the  two 
observers  are  2500  ft.  apart 
and  the  angles  of  elevation 
are  65°  20'  and  21°  32',  find 
the  height  of  the  airplane. 

10.  A  man  in  a  balloon  when  directly  over  a  road  con- 
necting two  towns  7  miles  apart  observes  the  angles  of 
depression  as  in  Fig.  117.     Find  the  height  of  the  balloon. 

Ans.  2.0361. 

11.  A  hill  has 
a  grade  of  9°.  If 
the  height  of  the 
hill  AB  is  59  ft., 
how  much  must 
be  taken  from  the 
top     of     the     hill 

Ans.  13.26  ft. 


Ans.  835.12. 


Fig.  117. 


to  make  the  grade  7°? 


CHAPTER  XIV 

TRIGONOMETRIC  FUNCTIONS  OF  ANY  ANGLE 

134.  Functions    as    Lines.     In    the    circle,    Fig.     118, 
let  the  radius  equal  1.     Then 


M 

/N 

D 

1                  ° 

/ 

C 

Fig.  118. 

AB    AB      .  D 
smx=m=-r=AB 

A0      in 

COS  X  =  jyr,  —  AU 

CD    rn 
tan  x=  jr^  =  CD 

MN 
cot  x(Zx=ZN)=-FlQ  =  MN 

Then,  if  the  line  OC  remains  fixed  and  the  line  OB 
revolves  about  0,  the  angle  x  increases,  but  sin  x  is  always 
the  perpendicular  distance  from  B  to  the  line  OC. 

164 


MEANING    OF    FUNCTIONS 


165 


Thus,  AB,  A'B',  A"B"  in  Fig.  119,  represent  the  sine  of 
the  different  values  of  x. 

From  Fig.  118  it  can  be  seen  that,  as  the  line  OB  revolves 
and  x  changes  from  0°  to  90°, 

sin  x  changes  from  0  to  1, 

cos  x  changes  from  1  to  0, 

tan  x  changes  from  0  to  cc  (oo  =  infinite  value), 

cot  x  changes  from  oo  to  0. 


\N 

M 

V 

M 

1   A               0 

^x         / 

Fig.  119. 


Fig.  120. 


135.  Value  of  the  Functions  of  Angles  Greater  than  90°. 

When  x  becomes  greater  than  90°,  Fig.  US  becomes  Fig.  120. 

,       .  _     '  ■'       adjacent  side    , 

Here    sine  x  cannot   be    denned  as   the    -r—  — ,  but 

hypotenuse 

is  defined,  on  a  broader  meaning  than  before,  as  the  per- 
pendicular distance  from  B  to  the  line  CO,  divided  by  the 
radius  of  the  circle  (  =  1).     That  is, 


AB  represents  sin  x 
OA  represents  cos  x 
CD  represents  tan  x 

M'N  represents  cot  x 


166     TRIGONOMETRIC   FUNCTIONS  OF  ANY  ANGLE 


136.  Signs  of  the  Functions  of  Angles  Greater  than  90°. 

As  x  increases  from  0°  to  90°,  cos  x  decreases  from  1  to  0°; 
but  as  x  keeps  on  increasing  from  90°  to  180°,  the  cosine 
keeps  changing  in  the  same  manner;  that  is,  it  decreases 
from  0  to  -1,  etc.  Hence,  in  Fig.  124,  the  diameters  must 
be  taken  as  coordinate  axes,  where  distances  to  the  right  are 
positive,  and  to  the  left  negative,  up  positive  and  down 
negative. 

From  Fig.  121  it  can  be  seen  that,  if  x  lies  between  90° 

and  180°: 

Y 


vN 

M 

K    /""" 

\\ 

\        \ 

A           \^ 

0          \ 

c 

^         J 

D 

w 

Fig.  121. 

sin  x  (  =  AB)   is  positive 
cos  x  (  =  AO)    is  negative 
tan  x  ( =  CD)    is  negative 
cot  x  ( =  MN)  is  negative 

If  x  lies  between  180°  and  270°,  as  in  Fig.  122: 

sin  x  (  =  AB)  is  negative 
cos  x  (  =  AO)  is  negative 
tan  x  ( =  CD)  is  positive 
cot  x  (  =  MN)  is  positive 


VALUES   OF   FUNCTIONS 


167 


The  signs  of  the  functions  of  the  angles  between  270c 
and  360°  can  be  worked  out  from  a  similar  figure.  Func- 
tions of  angles  greater  than  180°  are  very  seldom  used. 


Fig.  122. 


(corresponding    parts) . 


137.  Values  of  Functions  of  Angles  between  90°  and  360°. 
In  Fig.  123  triangles  OAB,  OA'B,  OA'B",  and  OAB'"  are 
congruent  (Geometry) . 

And    AB  =  A'B'  =  A'B"  =  AB'" 
That  is,  for 

Angles  between  90°  and  180°: 
sin  (180°  —  x)  =  sin  x 
cos  (180°  —  x)  =  —cos  x 
tan  (180°  —  x)  =  —tan  x 
cot  (180°  —  x)  =  —cot  x 

Angles  between  180°  and  270°: 
sin   (180°+a;)=-sin  x 
cos  (180°+rc)=  —  cos  x 

tan  (180°+:c)  =      tan  x  pIQ   123 

cot  (180°+z)=     cot  x 


168     TRIGONOMETRIC  FUNCTIONS  OF  ANY  ANGLE 

Angles  between  270°  and  360°: 
sin   (360°-.r)=-sin  x 
cos  (360°  —  x)  =     cos  x 
tan  (360°-z)  =  -tan  x 
cot  (360°— x)  =  —cot  x 

The  values  of  the  functions  of  angles  greater  than  90° 
can  also  be  computed  by  drawing  a  similar  figure  and 
finding  sin  (90° -\-x)  in  terms  of  x,  etc.  This  solution  will 
be  found  in  texts  which  are  more  complete  in  the  theoretical 
part  of  trigonometry. 

Example  1.     Find  sin  123°. 

sin  123°  =  sin  (180° -123°)=  sin  57°=  .83867. 

EXERCISE    1 

Find: 

1.  sin   100°.  Ans.         98481. 

2.  sin   170°.  .17365. 

3.  cos  160°.  -.93969. 

4.  tan  150°.  -.57735. 

5.  cot  120°.  -.57735. 

6.  sin  200°.  -.34202. 

7.  cot  240°.  .57735. 

8.  cos  300°.  .5. 

9.  tan  122°  45'.  -1.5547. 

10.  sin   145°  32'.  .56593. 

11.  cos  104°     7.  -.24390. 

12.  cot  125°  38'.  -.71681. 


CHAPTER  XV 

OBLIQUE  TRIANGLE 

138,  Law  of  Sines.     In  the  triangle  shown  in  Fig.  124, 
h  is  perpendicular  to  A B.     Then, 


•     a     h 
sin  A=T. 

o 


sin  B  = 


h 


a 

sin  A  _  a 
sin  B     b 


Similarly  it  can  be  shown  that: 

sin  A  _  a 
sin  C     c 


and 


sin  B  _  b 

gin  C     c 

169 


(1) 
(2) 


=1     Dividing  (1)  by  (2). 


170 


OBLIQUE   TRIANGLE 


By  rearranging  terms  in  the  above  equations  they  can  be 
put  in  the  form, 

a  b  c 

sin  A     sin  B     sin  C 

The  above  equations  are  known  as  the  law  of  sines  and 
should  be  memorized. 

Example  1.  Solve  the  triangle  of  Fig.  125  for  Z  C, 
and  for  x. 


ZC+42°+  75°=  180° 

ZC+117°=180° 

ZC=  63° 


sin  42°     sin  75° 

21 X  sin  42°  (The  computation  can  be  done 
most  efficiently  by  means  of 
logarithms) 


x  = 


sin  7o 
21 X. 66913 


X        .96592 
x=  14.54 

Example  2.  Solve  the  triangle 
of  Fig.  126  for  Z  B. 

45  38 

sin  B    sin  54°  30' 
.     D    45Xsin54°30' 

SmB=        ~3S ' 

sin  5  =  0.84464 
sin  57°  38' =  0.84464 
Z£  =  57°  38' 


LAW   OF   SINES 


171 


Note.  Two  triangles  are  possible  the  one  as  shown  in  heavy  lines, 
and  the  one  as  shown  in  dotted  lines.  The  value  of  Z.B  in  the  second 
triangle  is 

180° -57°  38'; 
since 

sin  B  =  sin  (180 -B). 


EXERCISE   1 

Find  the  unknown  parts  in  each  of  the  following  triangles: 
1.  Fig.  127.  Ans.    ZC=  112°  31'. 

C  z=12.913. 

AM  y  =  31. 593. 


Ans.  Z£  =  46°  36'. 
M  =  2362.6. 
iV=2051.5. 


2.  Fig.  128. 

123^27      " N 


3.  Fig.  129. 

630 


Ans.    Z5  =  37°  58'. 

ZA  =  65°    2'. 

d  =  929.48. 


Fig.  129. 


172  OBLIQUE  TRIANGLE 

139.  Law  of  Cosines.     In  Fig.  124, 

b2=~Wi+h2 
AD  =  c-DB 
AD2  =  c2-2cDB+DB2 

b2  =  c2  -  2cBB+BD2+h2 

DB2+K2  =  a2 

—  =  cos5;    DB  =  a  cos  B 
a 

Therefore : 

b2  =  c2+a2-2ac  cos  B 
Similarly, 

d2  =  b2+c2-2bc  cos  A 

and 

c2  =  a2  +  b2-2abcosC 

These  three  formulas  have  the  same  form  and  meaning 
and  are  known  as  the  law  of  cosines.  One  form  must  be 
memorized. 

Example  1.     Solve  the  triangle,  Fig.  130,  for  x. 

x2  =  g2  + 102  -  2  X  8  X 10  X  cos  78° 

x2-64  +  100-2X8XlOX  .20791 

8 

02=64+100-33.264 
32= 130.736 
a;  =  11.433 


LAW   OF   COSINES  173 

Example  2.     Solve  the  triangle  of  Fig.  131  for  angle  A. 
62  =  52-2X5X7  cos  A 
36  =  25+49-70  cos  A 
70  cos  A  =  38 
cos  A  =  .  5429 
ZA  =  57°  T 


Fig.  131. 


EXERCISE  2 
Solve   the  following  triangles: 
1.  Fig.  132. 


Ans.  z  =  5.2915. 


Fig.  132. 
2.  Fig.  133. 


Ans.  re  =14.987. 


174 


OBLIQUE   TRIANGLE 


3.  Fig.  134. 


Ans.    ZA  =  36°  52'. 


Fig.  134. 


140.  Law  of  Tangents.  In  some  cases  it  is  more  efficient 
to  use  a  relation  between  the  sides  and  angles  of  a  triangle 
known  as  the  law  of  tangents. 


a  +  b      tanh(A+B) 
Law  of  tangents:  £=$= -^i(a=BT 


See  Fig.  135. 


The  proof  is  omitted. 


Fig.  135. 


141.  Finding  an  Angle  When  Three  Sides  Are  Given. 
When  the  three  sides  of  a  triangle  are  given,  the  following 
formula  is  more  efficient  for  finding  an  angle : 


sin 


lA-^j 


\s-b)(s-c) 
be 


where  a,  b,  and  c  are  the  sides  of  the    triangle  and  s  = 

(a+b+c). 


AREAS   OF   TRIANGLES 


175 


142.  Areas  of  Triangles.  The  area  of  a  triangle  is  ex- 
pressed by  the  formula  S=^bh,  where  b  is  the  base  and  h 
is  the  altitude. 

When  two  sides  and  the  included  angle  are  given  (Fig.  136) : 

h      ■    n 
-  =  sin  C 


h  =  a  sin  C 

Therefore 

S  =  ^absinC  Fig.  136. 

When  two  angles  and  the  included  side  are  given  (Fig.  137) 
B 


a 

b 

sin 

A 

sin  B 

a= 

b  sin  .'1 

sin  B 


Substituting  in  the  formula  of  previous  method: 

1  b2  sin  A  sin  C 
S~2         sinB 

£=180-(A+C) 

sin  £  =  sin  (180-5)  =  sin  (A +C) 

Therefore, 


S  = 


1  b2  sin  A  sin  C 

2  sin  (A  +  C) 


Problems  falling  under  the  two  above  cases  can  be  solved 
also  by  use  of  the  law  of  sines  and  right  triangulation. 


176 


OBLIQUE   TRIANGLE 


When  three  sides  arc  given: 

The  following  formula  is  given  where  a,  b,  and  c  are 
the  three  sides  and  s^Ka+b  +  c), 

S=  Vs(s-a)(s-b)(s-c) 

The  proof  can  be  worked  from  the  law  of  cosines  and 
sin2  x+cos2  x=l. 

EXERCISE  3 

Find  the  area  of  the  following  triangles: 
1.  Fig.  138. 


5/ 

/go0 

Fig 

8 
.  138. 

2. 

Fig. 

139. 

Ans.  17.320. 


Ans.  240. 


3.  Fig.  140. 


16 
Fig.  140. 

Ans.  55.427. 


4.  Fig.  141. 


Fig.  141. 
Ans.  97.880. 


CHAPTER  XVI 

ELECTRICAL   APPLICATIONS 

143.  Projection.     The  -projection  of  one  line  upon  another 
line  is  the  portion  of  that  second  line  included  between  the 


.»^*2 

JB' 

E                            C 

D 

F 

Fig 

142. 

perpendiculars  dropped  upon  it  from  the  extremities  of  the 
first  line.     The  projection  of  AB  upon  EF  Fig.  142,  is  CD. 

The  projection  of  one  line  upon  another  is  found  by  multi- 


Fig.  143. 

plying  the  first  line  by  the  cosine  of  the  angle  between  the  two 
lines.     (Angle  1  is  the  angle  between  AB  and  EF.) 

177 


178 


Proof : 


ELECTRICAL  APPLICATIONS 

/o     AB' 

cos  Z.2 


AB 

AB  cos  Z2  =  AB' 

Z  1  =  Z  2  (Corresponding  angles) 
AB'  =  CD  (Parallels  between  parallels) 
AB  cos  Z1  =  CD  (Substitution) 

EXERCISE   1 

(Refer  to  Fig.  143) 

1.  IfAB=8",  Zl  =  35°,  find  CD.  Ans.  6.5536. 

2.  If  AB  =  U",  Z  1  =  75°,  find  CD.  Ans.  3.6232. 

3.  If  Z  1  =  30°,  how  long  must  AB  be  to  make  CD  =  10' 

Ans.   11.545. 

4.  If  AB=  12",  and  CD  =  9",  find  Z  1. 


5.  If  Z  1  =40°,  and  AB=  12",  find  .4/?. 


Note.      AE  =  BB'. 


Ans.  41°  24'. 


Ans. 


7136. 


Fig.  144. 


VOLTAGE  GENERATED  BY  A  ROTATING  LOOP  179 

144.  Lines  of  Force  Cut  by  a  Loop.  A  B,  Fig.  144,  repre- 
sents a  field  of  magnetic  flux  or  lines  of  magnetic  force. 
OCDM  is  a  loop  of  wire  rotating  about  the  line  OM .  The 
wires  OC,  MD,  and  OM  do  not  cut  any  lines  of  force  in  any 
position  of  rotation.  At  the  position  OC",  CD  is  not  cutting 
any  lines,  but  at  the  position  OC,  CD  is  cutting  some  lines, 
and  at  the  position  OC,  CD  is  cutting  lines  at  its  maximum 
rate. 

EXERCISE   2 

1.  Suppose  CD  in  the  position  shown  by  OC,  Fig.  144, 
is  cutting  lines  at  the  rate  of  120.  At  what  rate  will  it  be 
cutting  lines  when  Z  0  =  50°,  if  the  angular  velocity  remains 
the  same?  Ans.  91.92. 

2.  Taking  OC"  as  the  starting  position  and  E  as  the 
maximum  rate  of  cutting  lines  (position  OC)  find  the  formula 
for  the  rate  of  cutting  lines  at  any  position  OC,  in  terms  of 
4>  and  E.     Call  this  rate  e.  Ans.  e=E  sin  <£. 

145.  Voltage  Generated  by  a  Rotating  Loop.  A  voltage 
will  be  generated  in  the  loop  OCDM,  Fig.  144  proportional 
to  the  rate  at  which  CD  cuts  the  lines  of  force.  The  maxi- 
mum voltage  is  obtained  in  the  position  OC  (0  =  90°)  and 
the  minimum  voltage  in  the  position  0C"(<f)  =  Qo).  The 
voltage  at  any  position  OC  is  called  the  instantaneous 
voltage,  and  is  expressed  by  the  formula  derived  in  problem 
2  above. 

EXERCISE   3 

1.  If  the  maximum  voltage  is  10,  compute  the  instan- 
taneous voltage  for  every  10°  up  to  360°  Tabulate  the 
results. 

2.  Plot  a  curve  of  the  results  obtained  in  Problem  1 
with  the  values  of  4>  as  abscissas  and  the  instantaneous 


180 


ELECTRICAL  APPLICATIONS 


voltages  as  ordinates.     (This   is   a   sine   or  voltage  curve. 
Save  it.  for  future  use.) 

3.  From  the  curve  of  Problem  4  read  the  value  of  the 
voltage  at  45°,  135°,  225°  and  315°. 

Ans.  7.1,  7.1,  -7.1,  -7.1. 

146.  Phase  Angle.  The  value  of  the  instantaneous  volt- 
age can  be  shown  also  by  revolving  a  line,  of  a  length  repre- 
senting the  maximum  voltage,  about  a  point.  The  angle 
between  the  line  and  the  X-axis  is  the  angular  position  <j>, 
and  is  called  the  phase  angle.  The  projection  upon  the 
Y-axis  is  the  instantaneous  voltage  (  =  Esm4>).  The 
diagram  is  arranged  as  Fig.  145. 


Fig.  145. 


EXERCISE  4 

1  With  scale  and  protractor  compute  the  instantaneous 
voltage  if  E=  10,  when  0  =  25°,  80°,  120°  and  325°. 

Ans.  4.2,  9.8,  8.7,  -5.7. 

2.  On  the  same  axis  and  scale  as  used  in  Problem  2, 
Exercise  3,  plot  another  curve  where  #  =  8  and  </>  is  0  when 
the  4>  of  the  other  curve  is  30°.  (This  curve  will  represent 
a  voltage  curve  of  a  loop  similar  to  OCDM  but  30°  behind 
it  and  not  as  large  a  loop.) 

3.  If  the  two  loops  are  connected  in  series  and  rotated, 
the  combined  voltage  at  any  time  will  be  the  sum  of  the  two 


PHASE  ANGLE 


181 


300-Kilowatt  Three-Phase  Turbo-Alternator. 


35,000-Kilowatt  Three-Phase  Turbo-Alternator. 


182  ELECTRICAL  APPLICATIONS 

instantaneous  voltages.  Plot  a  curve  of  such  sums  on  the 
same  scale  and  axis  as  used  in  Problem  2.  (Use  the  com- 
pass or  dividers  to  lay  off  the  sums,  and  save  reading  the 
numerical  values.)  (This  curve  is  the  resultant  curve  of 
the  sum  of  the  other  two  curves.) 

Compare  the  maximum  of  the  resultant  curve  with  the 
sum  of  the  maximum  of  the  other  two  curves.  Can  you  see 
from  the  two  curves  why  these  two  quantities  are  not 
equal? 

147.  Vectors.  The  method  of  Problem  3  above  is  one- 
of  the  methods  of  adding  two  A.C.  voltages.  The  two 
voltages  can  also  be  added  by  vector  addition. 

Some  physical  quantities  can  be  represented  completely 
by  means  of  a  straight  line.  The  length  of  the  line  can 
be  used  to  represent  the  numerical  value  of  the  quantity 
and  the  direction  of  the  line  to  represent  the  sense  of  the 
quantity.  For  example,  a  distance  of  3  miles  can  be  repre- 
sented by  a  line  3"  long,  and,  if  it  is  a  distance  to  the  east, 
that  fact  can  be  represented  by  making  the  line  point  to 
to  the  right  by  means  of  an  arrow  head  thus: 

Then  a  line  5"  long  and  pointing  in  the  opposite  direction 
thus 


4- 


will  represent  a  distance  of  5  miles  to  the  west.  Such 
quantities  are  called  vector  quantities.  A  straight  line 
representing  a  vector  quantity  is  called  a  vector. 

148.  Scalar  Quantities.  Quantities  which  can  be  ex- 
pressed completely  by  stating  their  magnitude  are  called 
scalar  quantities.  An  example  of  a  scalar  quantity  is  5 
bushels  or  7  ohms,  etc. 


THE  USE  OF  VECTORS  183 

149.  The  Use  of  Vectors.  Many  quantities  in  electricity- 
can  be  expressed  by  vectors  much  more  clearly  than  by 
words.  Vectors  form  a  valuable  method  for  adding  and 
subtracting  certain  quantities,  and  are  used  in  all  texts 
on  electricity.  Therefore  it  is  necessary  to  study  methods 
of  expressing  quantities  by  vectors  and  interpreting  the 
meaning  of  quantities  expressed  by  vectors. 

Example.  A  man  travels  4  miles  east  and  then  3  miles 
north.  How  far  and  in  what  direction  is  he  from  the 
starting  point.     Solve  by  vectors. 

Solution. 

The  line  AC,  Fig.  146  represents  4  miles  to  the  east  and 
the  line  CB  represents  3  miles  to  the  north.     The  line  AB 
represents  the  single  equivalent 
journey.     AB  is  called  the  vector 
sum  of  AC   and  CB.     If  AB  is 
measured  to  the  same  scale  as 
AC  and  CB,  it  will  be  found  to 
represent    5    miles,    and   if   the 
angle  A  is  measured  with  a  pro- 
tractor it    will  be  found   to  be 
about  37°.    Therefore  the  vector 
AB  is  interpreted  as  represent- 
ing a  distance  of  5  miles  in  the  direction  of  37°   to   the 
north  of  east. 

EXERCISE  5 

Dolve  by  vectors : 

1.  A  man  travels  10  miles  south  then  12  miles  east. 
How  far  and  in  what  direction  is  he  from  the  starting  point? 

2.  A  man  travels  10  miles  west  then  2  miles  north  and 
8  miles  southeast.  How  far  and  in  what  direction  is  he 
from  the  starting  point.     . 


184 


PRACTICAL  APPLICATIONS 


150.  The  Composition  of  Forces.  Another  and  more 
important  use  made  of  vectors  is  to  study  the  effect  of  two 
forces  acting  upon  the  same  object. 

Example.  An  object  is  acted  upon  by  two  forces,  one 
of  6  lbs.  and  one  of  8  lbs.  acting  at  an  angle  of  90°,  Fig.  147. 
What  is  the  value  and  direction  of  the  single  force  that 
would  produce  the  same  effect? 


Fig.  147. 


Fig.  148. 


By  a  law  of  mechanics  a  force  produces  the  same  effect 
upon  an  object  whether  it  acts  independently  or  in  con- 
nection with  other  forces.  Therefore,  A B  and  CD  acting 
together  would  produce  the  same  effect  as  if  AC  acted  and 
then  AB  acted.  This  could  be  represented  as  in  Fig.  148. 
Then  AB  represents  the  value  and  direction  of  the  single 
resultant  force.  Or  taking  the  forces  as  shown  in  Fig.  147 
and  completing  a  parallelogram  as  Fig.  149  the  diagonal 
AD  also  represents  the  resultant.  AB  of  Fig.  148  =  AD 
of  Fig.  149,  since  CD  =  AB  (opposite  sides  of  a  parallelogram) . 


Fig.  149. 


SUBTRACTION  OF  VECTORS  185 

EXERCISE   6 

1.  Add  a  force  of  5  lbs.  to  a  force  of  12  lbs.  acting  at 
right  angles  to  it. 

2.  Add  a  force  of  8  lbs.  to  a  force  of  12  lbs.  acting  at  an 
angle  of  60°. 

Hint.  Draw  the  vectors  at  an  angle  of  60°  and  complete 
the  parallelogram,  etc. 

3.  Add  a  force  of  3  lbs.  acting  horizontally  and  a  force 
of  4  lbs  acting  at  an  angle  of  20°  above  horizontal,  and  a 
force  of  2  lbs.  acting  at  an  angle  of  50°  above  the 
horizontal. 

Hint.  Three  or  more  vectors  can  be  added  by  adding 
two  and  then  the  third  one  to  that  sum,  etc.,  or  they  may 
be  added  by  being  placed  end  on  end  in  the  proper  direction 
as  Fig.  150. 


jg0?^ 

/so" 

""^^    ^^^c 

>  ^-"-20° 

3 

Fig.  150. 

4.  Add  a  force  of  5  lbs.  acting  horizontally,  and  a  force 
of  6  lbs.  acting  at  an  angle  of  37°  above  the  horizontal  and 
a  force  of  4.5  lbs.  acting  at  an  angle  of  60°  above  the  hori- 
zontal. 

151.  Subtraction  of  Vectors.  From  algebra  it  is  known 
that  subtracting  a  number  is  the  same  as  adding  an  equal 
number  with  the  opposite  sign,  thus: 

4-(+2)=4+(-2). 


186 


ELECTRICAL  APPLICATIONS 


Similarly,  in  the  subtraction  of  vectors,  to  subtract  vector 
1  from  vector  2,  add  to  vector  2  a  vector  (3)  equal  to  vector 
1  but  of  opposite  direction,  as  Fig.  151. 


Fig.  151. 

Vector  4  then  is  the  difference  between  vectors  1  and  2. 

Subtraction  of  vectors  can  be  arranged  more  con- 
veniently as  in  Fig.  150.  Complete  the  parallelogram  for 
adding  vectors  1  and  2. 


Fig.  152. 


Vector  5  represents  the  sum  of  1  and  2.  Vector  6  is 
equal  and  parallel  to  vector  4,  and  therefore  represents 
the  difference  between  1  and  2. 


SUBTRACTION  OF  VECTORS  187 

Proof.      Vector  3  =  Vector  1  (Construction) 

Vector  3  =  line  7  (Opposite  sides  of  a  parallelo- 
gram) 
Therefore,     Vector  1  =  line  7 

Vector  1  is  parallel  to  line  7  (Const.) 
Therefore  lines  1,  4,  7,  6  form  a  parallelogram. 
Therefore  vector  6  is  equal  to,  and  parallel  to  vector  4, 
hence   vector  6   represents   the   vector   difference   between 
1  and  2. 

Rule.  When  two  vectors  are  drawn  from  a  point  and  the 
parallelogram  is  completed,  the  diagonal  of  the  parallelogram, 
drawn  from  the  point  is  the  vector  sum  of  the  two  vectors  and 
the  other  diagonal  is  the  vector  difference. 

Note.  The  vector  difference  may  be  directed  in  either  direction, 
depending  upon  the  order  in  which  the  vectors  are  subtracted. 

EXERCISE  7 

1 .  Add  by  vectors  a  force  of  40  lbs.  and  a  force  of  60  lbs. 
acting  at  an  angle  of  30°. 

Note.  Alternating  current  voltages  may  differ  in  phase,  that  is 
in  the  time  of  reaching  maximum  or  minimum  value.  Phase  is  ex- 
pressed as  an  angle  which  corresponds  to  the  difference  in  position  of 
two  coils  on  the  armature.  Alternating  current  voltages  can  bo  rep- 
resented by  vectors,  the  phase  angle  is  generally  measured  from  a 
horizontal  line  to  the  vector. 

2.  Add  by  vectors  a  voltage  of  8  volts,  phase  angle  15°, 
to  a  voltage  of  10  volts,  phase  angle  45°.  Compare  the 
sum  with  the  maximum  resultant  from  the  curves  of 
Problem  3,  Exercise  4. 

3.  Check  the  result  of  Problem  2  by  solving  the  tri- 
angle by  trigonometry. 


188  ELECTRICAL  APPLICATIONS 

4.  Add  by  vectors  12  volts,  phase  angle  10°,  15  volts, 
phase  angle  35°  and  8  volts,  phase  angle  20°.  Measure 
magnitude,  phase  angle,  and  instantaneous  value  of  the 
resultant.     (See  page  180.) 

5.  Solve  No.  4  by  trigonometry. 

6.  Study  the  vector  diagram  of  Problem  4  to  see  what 
can  be  read  from  it  in  regard  to  instantaneous  values  of 
the  separate  voltages,  etc. 

7.  Circuits  of  12  volts,  phase  angle  20°  and  20  volts, 
phase  angle  45°  are  connected  in  series.  Find  the  vector 
sum  and  difference. 

8.  A  circuit  has  110  volts,  phase  angle  36°.  It  is 
desired  to  make  the  circuit  contain  150  volts,  phase  angle 
2.8°.     How  many  volts  must  be  added  and  in  what  phase. 

9.  Solve  No.  8  by  trigonometry. 

152.  Current.  Current  is  proportional  to  voltage,  there- 
fore, the  current  curve  is  similar  to  the  voltage  curve  and 
is  expressed  by  the  formula 

i=I  sin  </>, 

where  i  is  the  instantaneous  current  and  /  the  maximum 

current. 

If  the  circuit  contains  no  inductance  or  capacity,  the 
current  will  be  in  phase  with  the  voltage.  That  is,  the  cur- 
rent will  be  zero  when  the  voltage  is  zero,  and  maximum 
when  the  voltage  is  maximum. 

If  the  circuit  contains  inductance  only,  the  current  will 
lag  90°  behind  the  vo  tage. 

If  the  circuit  contains  capacity  only,  the  current  will 
lead  the  voltage  by  90°.  Most  alternating  current  circuits 
contain  resistance,  inductance  and  capacity.  The  current  is 
not  ordinarily  in  phase  with  the  voltage. 


POWER 


189 


EXERCISE  8 

1.  Draw  roughly  a  current  curve  in  phase  with  its  volt- 
age curve. 

2.  Draw   roughly   a   current   curve  leading  its  voltage 
curve  by  90°. 

3.  Draw   roughly   a   current   curve   lagging   behind   its 
voltage  curve  by  90°. 

4.  Draw   roughly   a   current   curve   lagging   behind   its 
voltage  curve  by  30°. 

The  above  exercises  can  be  represented  vectorially  as 
in  Fig.  153. 


(a) 


(c) 


(b) 


Fig.  153. 


(d) 


Fig.  153  (a)  represents  current  in  phase  with  the  voltage. 
Fig.  153  (6)  represents  the  current  leading  the  voltage  by 
90°.  Fig.  153  (c)  represents  the  current  lagging  90°  behind 
the  voltage.  Fig.  153  (d)  represents  the  current  lagging 
30°  behind  the  voltage. 

153.  Power.  From  the  curves  of  Exercise  8,  it  is  evident 
that,  at  any  instant,  the  power  is  equal  to  the  instantaneous 
current  times  the  instantaneous  voltage. 


190 


ELECTRICAL  APPLICATIONS 


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Double  Circuit  Three-Phase  Transmission  Line  with  Pin  Type  Insulators. 


POWER  191 

The  current  and  voltage  do  not  reach  a  maximum  at 
the   same  time,   hence   maximum   power  is   not   maximum 
current  times  maximum  volt- 
age.   Similarly,  effective  power  -  //  \ 

is  not  equal  to  effective  cur-  J^ 

rent    times  effective  voltage,      /     ^ j 

but,   the  power  at  any  instant  Ecos<£  i 

is   equal    to    the  current  times  Fig.  154. 

the   component   of    the    voltage 

that  is  in  the  same  phase  with  the  current.     Graphically,  this 

can  be  represented  as  in  Fig.  154. 

Power,  then,  is  represented  by  the  formula 

P  =  EI  cos  <t>, 

where  P  is  the  maximum  of  effective  power,  E  is  the  maxi- 
mum or  effective  voltage  and  7  is  maximum  or  effective 
current.  4>  is  the  phase  angle  between  the  current  and  volt- 
age, and  cos  <f>  is  called  the  power  factor. 

EXERCISE   9 

1.  If  r=10.5,  #=125,  and  0  =  40°  find  the  power. 

2.  In  Problem  1  what  is  the  power  factor? 

Ans.  .766a. 

3.  In  an  A.C.  circuit  the  current  lags  40°,  the  volt- 
meter indicates  120  volts,  the  ammeter  16  amperes.  What 
would  a  wattmeter  read? 

4.  An  ammeter  in  a  120  volt  A.C.  circuit  indicates  8 
amperes.  A  wattmeter  in  the  same  circuit  reads  500  watts. 
Find  the  power  factor  and  phase  difference. 

5.  A  110  volt  A.C.  generator  is  to  deliver  2100  watts 
to  a  circuit  having  a  power  factor  of  0.84.  What  current 
is  necessary? 


192  ELECTRICAL  APPLICATIONS 

154.  Force  on  a  Conductor  Carrying  Current  in  a  Mag- 
netic Field.  The  force  exerted  on  a  conductor  carrying 
current  and  lying  in  a  magnetic  field  is  expressed  by  the 

equation 

IIH  sin  4> 


F  = 


10 


where    F  =  force  in  dynes  on  the  conductor ; 
/  =  current  through  the  conductor; 
1  =  length  of  the  conductor  in  centimeters; 
H  =  strength  of  the  magnetic  field  in  gausses, 
<j>  =  angle  between  the  conductor  and  the  magnetic 
field. 

EXERCISE   10 

1.  Find  the  force  on  a  conductor  45  cm.  long  carrying 
a  current  of  12  amperes  in  a  field  of  16,000  gausses,  if  the 
angle  between  the  conductor  and  the  lines  of  magnetic 
forcois80°.  Ans.  850,860. 

2.  How  great  a  magnetic  flux  will  be  required  to  exert 
a  maximum  force  (0  =  90°)  of  1,250,000  dynes  on  a  conductor 
18"  long  carrying  a  current  of  5  amperes?       Ans.  54,680. 

3.  Find  <f>  when  F  =  2,350,000,  J =8,  1=2',  H=  125,000. 

Ans.  22°  40'. 

155.  Impedance.  Impedance  in  an  alternating  current 
circuit  may  be  computed  from  the  formula 


Z^+^L-^f, 


where    Z  =  impedance ; 

R  =  resistance  of  circuit  in  ohms; 
/=  number  of  cycles  per  second; 
L  =  inductance  in  henries; 
C  =  capacity  in  farads. 


FORCE  ON  A  CONDUCTOR  193 


Double  Circuit  Three-Phase  Transmission  Line  with  Suspension  Insulators. 


194  ELECTRICAL  APPLICATIONS 

EXERCISE   11 

1    Compute  Z  when  R  =  230,  /=  60,  L  =  .15,  C  =  .0003. 

Ans.  235.1. 

2.  When  L  =  .122  and  C  =  . 00024,  what  value  of  /  will 

make  (2ir/L- ^)  =  °  ?  Ans-  29"4L 

3.  When  C  =  .0004  farad  and  /=500,  what  value   of  L 

will  make  (tofL  -  ^)  =  0  ?  Ans.  .00025. 

Note.     High  frequency  currents  are  used  in  radio  work. 
156.  Computation    of    Current.     Current    in    an    alter- 
nating current  circuit  may  be  found  from  the  equation 

E 


1  = 

where    J  =  current ; 
E  =  voltage. 


^2+(2^-2?c)2' 


EXERCISE   12 


1.  Find  the  current  that  will  flow  through  a  circuit  of 
125  ohms,  .02  henry,  and  .00003  farad,  when  a  60-cycle  120- 
volt  alternating-current  voltage  is  applied.  Ans.   .806. 


MISCELLANEOUS  PROBLEMS 

The  problems  which  follow  are  selected  to  give  to  the 
student,  as  nearly  as  possible,  practical  experience  in  com- 
putation. The  problems  are  not  classified  but  cover  a  wide 
and  general  field.  Logarithms  should  be  used  in  the 
computations. 

1.  A  distance  AB  of  154.59',  Fig.  155,  is  measured  along 
the  bank  of  a  river.  A  tree  C  is  sighted  on  the  opposite  bank, 
making  the  angle  at  A  equal  to  34°  28',  and  the  angle  at  B 
equal  to  90°.     Find  the  width  of  the  river  BC. 

Ans.  106.11  ft. 


34°28' 


Fig.  155. 


Fig.  156. 


2.  To  find  the  distance  between  A  and  B  separated  by 
a  lake,  Fig.  156,  readings  were  made  as  shown. 

Find  the  distance  AB.  Ans.  952.4. 

195 


196 


MISCELLANEOUS   PROBLEMS 


3.  Two  sides  of  a  triangular  lot  are  found  to  be  173  rods 
and  194  rods.  The  included  angle  is  56°  15'.  Find  the 
area.  •  Ans.  87 . 2  acres. 

4.  Eight  holes  are  to  be  drilled  in  a  circle  of  radius  5  in. 
How  far  apart  are  the  holes?  Ans.  3.8267. 

5.  To  drill  the  holes  in  the  above  problem  it  will  be 
necessary  to  move  the  table,  as  in  Fig.   157. 


D   F 


Fig.  157. 


Find 

Ans. 

AB. 

1.464 

BC. 

3.536 

AM. 

5. 

MN. 

5. 

AD. 

8.537 

DE. 

3.536 

AF. 

10. 

6.  The  latitude  of  Washington,  D.  C,  is  38°  55'  N. 
How  many  miles  east  and  west  on  the  earth's  surface  make 
a  difference  in  time  of  1  hour  in  the  latitude  of  Washington? 
Radius  of  the  earth  is  3957  miles.  Ans.  806 .  02. 

7.  Find  x,  Fig.  158. 


-    Ans.  0.0625. 


Fig.  158. 


MISCELLANEOUS   PROBLEMS 


197 


8.  Find  x,  Fig.  159  (angles  are  90°).  Ans.  4.0588. 


Fig.  159. 

9.  The  angle  of  elevation  of  the  top  of  a  mountain, 
observed  from  a  point  directly  south  of  it  is  60°.  From  a 
point  1  mile  directly  east  of  the  first  point  and  on  the  same 
level  with  it  the  angle  of  elevation  is  45°.  Find  the  height 
of  the  mountain.  Ans.  6466 . 8. 

10.  A  light-house  was  observed  to  bear  directly  east 
from  a  ship.  After  the  ship  sailed  4 
miles  north  the  light-house  bore  55°  30' 
east  of  south.  How  far  was  the  ship 
from  the  light-house  at  the  time  of  each 
observation?        Ans.  5.8202,  7.062. 

11.  One  side  of  a  hexagon  is  2  in. 
Find  the  area.  Ans.  10.3926. 

12.  Find  the  radius  of  the  circle,  Fig. 
160.  Ans.   .5429. 


Fig.  160. 


320  lbs 


410  lbs. 

Fig   161. 


13.  If  two  forces 
of  410  lbs.  and  320 
lbs.  pull  at  an  angle 
of  51°  37',  as  shown 
in  Fig.  161,  the  line 
AB,  which  is  the  di- 
agonal  of   the   paral- 


198 


MISCELLANEOUS   PROBLEMS 


lelogram  of  which  AM  and  AN  are  the  sides,  represents 
the  value  and  direction  of  the  resultant  force.  Find  the 
value  of  the  resultant  force  and  the  angle  which  it  makes 
with  the  410-lb.  force.  Ans.  658.36  lbs.,  22°  23'  47". 

14.  An  unknown  force  combined  with  a  force  of  128  lbs. 
produces  a  resultant  of  200  lbs.  and  this  resultant  makes 
an  angle  of  18°  24'  with  the  128-lb.  force.  Find  the  inten- 
sity and  direction  of  the  unknown  force. 

Ans.  88.326  lbs.,  54°  37'  16". 

15.  A. 


Find  the  tension  in   the 
cable  AB,  Fig.  162. 

Ans.  3901  lbs. 

Q  5000  lbs. 
Fig.  162. 
Note.     The  parallelograms  of  forces,  Fig.  163,  can  be  used. 


16. 


Find  the  distance  x 
between  the  plugs, 
Fig.  164. 

Ans.  2.2930  in. 


Fig.  164. 


MISCELLANEOUS   PROBLEMS 


199 


17. 


Find  the  distance 
Ans.  3.9065. 


Fig.  165. 


18.  From  the  top  of  a  hill  the  angles  of  depression  of 
two  objects  5280  ft.  apart  on  a  straight  level  road  leading 
to  the  hill  are  5°  and  15°.     Find  the  height  of  the  hill. 

Ans.  685.9  ft, 


APPENDIX 


DICTIONARY    OF    THE    TERMS    USED    IN 
ELEMENTARY    MATHEMATICS 

Abscissa.     The  horizontal  or  X  coordinate  of   a  point- 
Acute  Angle.     An  angle  of  less  than  90°. 
Adjacent.     Next  to  or  adjoining. 
Adjacent  Angles.     Angles  having   a   common   side   and    a 

common  vertex. 
Altitude.     Height  measured  perpendicular  to  the  base. 
Angle.     The  difference  in  direction  of  two  lines  that  meet 

at  a  point- 
Angle  of  Depression.     The  angle  made  by  the  line  of  sight 

to  an  object  below  the  horizontal  plane  of  the  observer, 

and  the  plane. 
Angle  of  Elevation.     The  angle  made  by  the  line  of  sight  to 

an  object  above  the  horizontal  plane  of  the  observer, 

and  the  plane. 
Antecedent.     The  first  or  third  term  of  a  proportion. 
Antilogarithm.     The  number  corresponding  to  a  given  loga- 
rithm. 
Apothem.     The  perpendicular  distance  from   the   center  of 

a  regular  polygon  to  one  of  the  sides. 
Arc.     Part  of  the  circumference  of  a  circle. 
Axiom.     A  truth  that  is  self-evident- 
Base.     The  side  or  surface  upon  which  a  geometrical  figure 

appears  to  rest. 
Base.     An  expression  which  is  to  be  raised  to  a  power. 
Binomial.     An  expression  having  two  terms. 
Bisector.     A    line    or   plane   which    divides    a   geometrical 

figure  into  two  equal  parts. 
201 


202  DICTIONARY   OF  THE   TERMS   USED 

Central  Angle.     An  angle  which  has  its  vertex  at  the  center 

of  a  circle. 
Characteristic.     The  whole  number  part  of  a  logarithm. 
Check.     Proof  of  the  solution  of  a  problem. 
Chord.     A  straight  line  connecting  two  points  on  the  cir- 
cumference of  a  circle. 
Circle.     A  plane   figure  bounded  by  a  curved   line   every 

point  of  which  is  equidistant  from  a  point  called  the 

center. 
Circular  Mil.     A  unit  of  area  equal  to  the  area  of  a  circle 

one  mil  in  diameter. 
Circumference.     The  curved  line  forming  a  circle. 
Circumscribed.     Drawn  about,  as  a  polygon  circumscribed 

about  a  circle. 
Coefficient.     The  coefficient  of  a  factor  is  the  product  of  all 

the  remaining  factors. 
Cologarithm.     The  logarithm  of  the  reciprocal  of  a  number. 
Commensurable    Quantities.       Quantities  whose   common 

unit  of  measure  is  a  rational  quantity. 
Complement.     The  angle  which  will  add  to  a  given  angle  to 

make  90°. 
Concentric.     Having  the  same  centers. 
Cone.     A  solid  generated  by  the  rotation  of  a  right  triangle 

about  one  of  its  legs. 
Congruent.     Equal  in  all  respects. 

Consequent.     The  second  or  fourth  term  of  a  proportion. 
Constant.     A  quantity  whose  value  does  not  change. 
Continued  Proportion.     A  proportion  of  three  or  more  ratios. 
Converse.     Reversed  in  the  order  of  relation. 
Coordinate.     The  distance  of  a  point  from  the  coordinate 

axes. 
Coordinate  Axes.     The  reference  lines  by  which  a  graph  is 

plotted. 


DICTIONARY  OF  THE  TERMS  USED  203 

Corollary.     A  truth  which  follows  naturally  from  another 

truth. 
Corresponding  Parts.     Parts  similarly  placed  in  congruent 

or  similar  geometric  figures. 
Cube.     A  rectangular  solid  having  all  of  its  faces  squares. 
Cube.     The  third  power  of  a  number. 
Cutting  Speed.     The  speed  at  which  work  passes  the  point 

of  a  cutting  tool. 
Cylinder.     The   solid   generated   by   revolving   a   rectangle 

about  one  of  its  sides. 
Definite  Number.     A  number  having  always  the  same  value. 
Degree.     3^0  of  the  angular  space  about  a  point. 
Diagonal.     A  line  connecting  any  two  non-adjacent  vertices 

of  polygon  or  polyhedron. 
Diameter.     A  line  passing  through  the  center  of  a  circle  and 

ending  at  the  circumference. 
Dihedral   Angle.     The   angle   formed   by   two   intersecting 

planes. 
Ellipse.     Section  of  a  cone  made  by  a  plane  cutting  the  cone 

not  parallel  to  or  cutting  the  base. 
Equation.     A  statement  that  two  expressions  are  equal. 
Equiangular.     A  plane  figure  having  equal  angles. 
Equilateral.     A  plane  figure  having  equal  sides. 
Equivalent.     Equal  in  area  or  volume. 
Evaluation.     The  process  of  substituting  definite  numbers 

for   general   numbers   and   performing   the   operations 

indicated. 
Exponent.     A  small  number  written  above  and  to  the  right 

of  a  number  (called  the  base)  to  show  how  many  times 

the  base  is  to  be  used  as  a  factor. 
Exterior  Angle.     The  angle  between  one  side  of  a  polygon 

and  an  adjacent  side  extended. 
Extremes.     The  first  and  fourth  terms  of  a  proportion. 


204  DICTIONARY   OF  THE  TERMS   USED 

Factor.     One  of  the  quantities  which,   multiplied   together, 

form  a  given  product. 
Formula.     Statement   of   a   rule   or   principle   in   terms   of 

general  numbers. 
Frustum.     The  part  of  a  cone  or  a  pyramid  next  the  base, 

formed  by  cutting  off  the  top. 
Fulcrum.     The  point  about  which  a  lever  turns. 
Function.     A  quantity  that  depends  upon  another  quantity 

for  its  value. 
Function  of  an  Angle.     Sin,  cos,,  or  tan,  etc.,  of  the  angle. 
General  Number.     A  number  that  has  different  values  in 

different  problems. 
Graph.     A    curve   representing   the   relation    between   two 

variables. 
Great  Circle.     The  largest  circle  that  can  be  drawn  on  the 

surface  of  a  sphere. 
Hexagon.     A  plane  figure  having  six  sides. 
Hypotenuse.     The  side  of  a  right  triangle  opposite  the  right 

angle. 
Incommensurable   Quantities.     Quantities  whose  common 

unit  of  measure  is  an  irrational  quantity. 
Index.     The  number  above  and  to  the  left  of  the  radical 

sign  which  indicates  what  root  is  to  be  taken. 
Index.     The  end  of  a  slide  rule  scale. 
Inscribed.     Drawn  within;  a  polygon  is  inscribed  in  a  circle 

when  all  the  vertices  of  the  polygon  lie  on  the  circum- 
ference of  the  circle. 
Inscribed  Angle.     An  angle  whose  vertex  lies  on  circumfer- 
ence of  a  circle  and  whose  sides  are  chords  of  the  circle. 
Intercept.     To  cut. 

Intercept.     The  part  included  between  two  points. 
Interpolation.     The  process  of  finding  intermediate  terms 

from  a  table  by  means  of  the  two  nearest  terms  given. 


DICTIONARY  OF  THE  TERMS   USED  205 

Irrational   Quantity.     A  quantity  whose  value  cannot  be 

expressed  exactly  by  decimals  or  fractions,  as  y/2. 
Isosceles  Triangle.  A  triangle  having  two  sides  equal. 
Least  Common  Denominator.      The  smallest  number  that 

will  contain  two  or  more  denominators  evenly. 
Lever.     A  rigid  piece  capable  of  turning  about  a  point  called 

a  fulcrum. 
Leverage.     Tendency  to  turn,  turning  moment. 
Lever  Arm.     Distance  from  the  fulcrum  of  a  lever  to  a  force 

acting  upon  the  lever,  measured  perpendicular  to  the 

direction  of  the  force. 
Locus.     The  path  of  a  point  or  curve  moving  according  to 

some  law. 
Logarithm.     The  exponent  of  that  power  of  a  base  (usually 

10)  which  equals  a  given  number. 
Lune.     A  portion  of  the  surface  of  a  sphere  bounded  by  the 

semicircumferences  of  two  great  circles. 
Mantissa.     The  decimal  part  of  a  logarithm. 
Mean  Proportion.     When  the  second  and  third  terms  of  a 

proportion  are  the  same  the  proportion  is  a  mean  pro- 
portion. 
Mean  Proportional.     The  second  (or  third)  term  of  a  mean 

proportion. 
Means.     The  second  and  third  terms  of  a  proportion. 
Median.     A  straight  line  drawn  from  the  vertex  of  a  triangle 

to  the  midpoint  of  the  opposite  side. 
Member.     One  of  the  parts  of  an  equation  separated  by  the 

equal  sign. 
Mil.     One  thousandth  of  an  inch. 
Monomial.     An  expression  having  only  one  term. 
Negative     Number.      A     number     to     be     subtracted,    or 

a  number    on   the   number   scale   to    the    left  of   the 

zero. 


206  DICTIONARY   OF   THE   TERMS   USED 

Oblique  Angle.  Any  plane  angle  not  a  right  angle,  straight 
angle,  or  perigon. 

Oblique  Triangle.     A  triangle  having  all  oblique  angles. 

Obtuse  Angle.  An  angle  greater  than  a  right  angle  and 
less  than  a  straight  angle. 

Octagon.     A  plane  figure  having  eight  sides. 

Ordinate.     The  vertical  or  Y  coordinate  of  a  point. 

Parallel  Lines.  Straight  lines  in  the  same  plane  that  will 
not  meet  if  extended. 

Parallelogram.  A  quadrilateral  having  its  opposite  sides 
parallel. 

Parallelopiped.  A  six-sided  solid  all  of  whose  faces  are 
parallelograms. 

Pentagon.     A  plane  figure  having  five  sides. 

Perigon.     An  angle  of  360°. 

Perimeter.     The  distance  around  a  plane  figure. 

Perpendicular.     At  right  angles  to  a  line  or  surface. 

Phase.     Position  in  a  cycle  of  rotation  or  oscillation. 

Place.     The  position  of  a  figure  in  a  number. 

Plane.     A  surface  without  curvature. 

Polygon.     A  plane  figure  bounded  by  straight  lines. 

Polyhedral  Angle.  The  solid  angle  formed  by  three  cr 
more  planes  meeting  at  a  point. 

Polyhedron.     A  solid  bounded  by  planes. 

Polynomial.     An  expression  having  two  or  more  terms. 

Positive  Number.     Ordinary  or  arithmetical  number. 

Power.     Product  of  equal  factors. 

Prism.  A  solid  whose  bases  are  congruent  and  parallel  and 
whose  sides  are  parallelograms. 

Projection.  (Of  a  point  on  a  line.)  The  foot  of  the  per- 
pendicular from  that  point  to  the  line.  (Of  a  line  or 
surface  on  a  line  or  surface.)     The  line  or  surface"  join- 


DICTIONARY  OF  THE   TERMS   USED  207 

ing  the   projection  of  all  points  of  that  line  or  surface, 

on  the  second  line  or  surface. 
Proportion.     When  two  ratios  are  equal,  they  form  a  propor- 
tion. 
Protractor.     An    instrument    for    drawing    and    measuring 

angles- 
Pyramid.     A  solid  having  for  its  base  a  polygon  and  for  its 

sides  triangles  meeting  at  a  common  point. 
Quadrant.     One-fourth  of  a  circle  or  perigon. 
Quadrilateral.     A  plane  figure  having  four  sides. 
Radius.     A  straight  line  from  the  center  of  a   circle  to  the 

circumference,  or  from  the  center  of  a  sphere  to  the 

surface. 
Ratio.     The  relation  of  one  quantity  to  another  quantity 

of  the  same  kind. 
Rational   Quantity.     A  quantity  whose  value  can  be  ex- 
pressed exactly  by  decimals  or  fractions. 
Reciprocal.     One  divided  by  a  given  number. 
Rectangle.     A  parallelogram  having  right  angles. 
Rectilinear.     Having  straight  lines. 
Reflex  Angle.     An  angle  greater  than  180°. 
Regular  Polygon.     A  polygon  that  has  its  sides  equal  and  its 

angles  equal. 
Resultant.     The  single  force  replacing  two  or  more  forces. 

Also  applies  to  other  vectors. 
Rhomboid.     A  parallelogram  having  its  angles  oblique  and 

its  adjacent  sides  unequal. 
Rhombus.     A  parallelogram  having  its  angles  oblique  and  its 

sides  equal. 
Right  Angle.     An  angle  of  90°. 
Right  Prism.     A  prism  whose  sides  are  perpendicular  to  the 

base. 


208  DICTIONARY   OF  THE  TERMS   USED 

Right  Triangle.     A  triangle  having  one  right  angle. 

Rim  Speed.  The  speed  of  a  point  on  the  surface  of  a  revolv- 
ing object. 

Runner.     The  sliding  glass  piece  on  a  slide  rule. 

Scalar  Quantity.  A  quantity  which  can  be  expressed  com- 
pletely by  giving  its  magnitude.  Sometimes  called  a 
scalar. 

Scalene  Triangle.  A  triangle  having  no  two  sides 
equal. 

Secant.     A  straight  line  passing  through  a  circle. 

Sector.     The  portion  of  a  circle  between  two  radii. 

Segment.  The  portion  of  a  circle  between  a  chord  and  the 
circumference. 

Similar.     Of  the  same  shape. 

Slide.     The  central  portion  of  a  slide  rule. 

Slide  Rule.     A  calculating  instrument. 

Solid.     Any  object  having  definite  shape. 

Sphere.  A  solid  bounded  by  a  curved  surface  every  point 
of  which  is  equidistant  from  the  center. 

Square.     The  second  power  of  a  number. 

Square.     A  rectangle  having  equal  sides. 

Square  Root.  The  number  which  when  multiplied  by  itself 
equals  a  given  number. 

Straight  Angle.     An  angle  of  180°. 

Substitution.  The  process  of  replacing  one  quantity  by 
another. 

Subtend.     Cut  off. 

Supplement.  The  angle  which  added  to  a  given  angle  will 
make  180°. 

Symmetrical.  The  same  on  both  sides  of  a  point  line,  or 
plane. 

Tangent.     Touching  but  not  cutting. 

Tetrahedron.     A  polyhedron  having  four  equal  faces. 


DICTIONARY   OF  THE  TERMS   USED  209 

Term.  An  expression  whose  parts  are  not  separated  by  the 
plus  or  minus  sign. 

Theorem.     A  statement  to  be  proved. 

Torque.     Forces  tending  to  produce  rotation. 

Transpose.  To  change  a  term  from  one  member  of  an 
equation  to  the  other. 

Transversal.     A  line  cutting  other  lines. 

Trapezium.     A  quadrilateral  having  no  two  sides  parallel. 

Trapezoid.  A  quadrilateral  having  one  pair  of  parallel 
sides. 

Triangle.     A  plane  figure  having  three  sides. 

Variable.     A  quantity  that  changes  in  value. 

Vector.     A  line  representing  a  vector  quantity. 

Vector  Quantity.  A  quantity  having  magnitude  and  direc- 
tion or  sense. 

Vertical  Angles.  If  two  lines  intersect,  the  opposite  angles 
are  vertical  angles. 


RELATIONS  BETWEEN  THE  TRIGONOMETRIC 
FUNCTIONS 

Certain  relations  exist  between  the  functions  of  angles. 
These  relations  are  used  in  finding  one  function  from  other 
functions.  The  following  table  is  given  for  reference. 
The  proofs  of  the  relations  are  given  in  books  more  com- 
plete in  the  theory  of  trigonometry: 

1.  sin2  z+cos2  x=l. 

„    sin  x 

=  tan  x. 


cos  a: 
cos  x 


=  cot  x. 


sin  x 

4. 

sec2  a;=l+tan2  x, 

5. 

esc2  £=1-1- cot2  x. 

6. 

1 

sin  x= -. 

7.  cos  x = 

8.  tan  x  = 


cscx 

1 

sees" 

1 


cot  X 

9.  sin  (x-\-y)  =sin  x  cos  y+cos  x  sin  y. 
10.  cos  (x+y)  =  cos  x  cos  y— sin  x  sin  y. 
tan  rc+tan  y 


11.  tan  (x+?/)  = 


1  — tanx  tan  ?/' 
211 


212     RELATIONS  OF  TRIGONOMETRIC  FUNCTIONS 

.     cot  x  cot  y  —  1 

12.  cot  (x-\-y)  =  — i — • 

v       J        cota;+cot  y 

13.  sin  (x  — y)  =  sin  x  cos  y  —cos  a:  sin  ?/. 

14.  cos  (x— y)=  cos  z  cos  2/+sin  x  sin  y„ 

.        .       tan  re— tan  i/ 

15.  tan  (x  —  y)  = ,— rr     — * • 

v      *"     1+tanrc  tany 

.        .     cot  x  cot  ?y  —  1 

16.  cot  (*-v)«-^+^t7' 

17.  sin  2a;  =  2  sin  a;  cos  a:. 

18.  cos  2a:  =  cos2  x  — sin2  jr. 

2  tan  a; 

19.  tan  2a;  =  y^Fx 

cot2  a;— 1 

20.  cot  2a;  =  -75 — t — • 

2  cot  x 


.    re            /l  — cos  re 
21.  sin?2=±^ 2^~- 


I +cos  re 
22.  cos 


re  /l  —  cos  a; 

23.  tang-i^jq^^. 

re  /l+cosre 

24.  cot2=±Vl^cos"x- 


FORMULAS 

The  following  list  of  formulas  have  a  wide  and  general 
use.  The  student  should  know  each  formula  to  have  a 
useful  knowledge  of  mathematics. 

1.  tt  =  3.1416. 

2.  C  =  2irr.  C  =  circumference  of  a  circle, 

r  =  radius. 

3.  C  =  — ~ — .     C  =  circumference  of  ellipse, 

d\  and  (h  =  long  and  short  diameters. 
Areas: 

4.  Square  A  =  a2.        a  =  side. 

5.  Rectangle  A  =  ab.       a  and  b  are  the  dimensions. 

6.  Parallelogram  A  =  bh.  b  =  base,  h  =  height. 

7.  Trapezoid  A  =  %h(b+b') .     /i  =  height.      b  and  b'  are 

the  parallel  sides. 

8.  Triangle  A  =  \bh.       6  =  base,     h  =  height. 

9.  Triangle  A  =  Vs(s  —  a)(s— b)(s— c). 

a,  b  and  c  are  the  sides,  s  =  %(a+b+c). 

10.  Hexagon  A  =  2.598s2.     s  =  side. 

11.  Circle  A=rr2.         r  =  radius. 

12.  Ellipse  A=ir—T— .     d\  and  rf2  =  long  and  short  diam- 

eters. 

13.  Surface  of  sphere  A  =4tt2.        r  =  radius. 
Volumes: 

14.  Cube  V  =  a3.  a  =  edge. 

15.  Rectangular  solid  V  =  lwh,  I,  w,  and  h  are  the  dimen- 

sions. 

213 


214  FORMULAS 

16.  Rectangular  solid,  Parallelopiped,  Prism  or  Cylinder. 
V  =  bh,  6  =  area  of  the  base,  ft  =  height. 

17.  Cone  or  Pyramid  V  =  \bh,  b  =  area  of  base,  ft  =  height. 

18.  Frustum  of  a  Cone  or  Pyramid. 

y  =  xjl(0-\-b'  +  Vbb'),  6  and  b'  are  the  areas  of  the 
bases  and  ft  =  height. 

19.  Sphere  F  =  ^rr\         r  =  radius. 

20.  Spherical  segment  V=  \ira [  r2+ —  I . 

a  =  altitude,     r  =  radius  of  the  base. 
Right  Triangle: 

21.  c2  =  a2-\-lr.     c  =  hypotenuse,  a  and  b  the  sides. 
Q uadratic  Equation : 

-b±\/b2+4ac 
22-  X  = 2aT     -' 

Trigonometry : 

<•    ■  a  b  c 

23.  Law  oi  sines: 


sin  A     sin  B     sin  C 
24.  Law  of  cosines  a2  =  b2  +  c2  —  2bc  cos  A. 


TABLE    1.— DECIMAL   EQUIVALENTS 


Of  Eighths,  Sixteenths,  Thirty-seconds  \\n  Sixty-fourths  of  an 

Inch 


.015625 
.03*125 

.046875 

.0625 

.078125 

•09375 

•109375 

.1250 

. 140625 

■15625 

•171875 

•  1875 
.203125 
.21X75 

•  234375 
.2500 
.265625 
.28125 
.296875 
■  3125 
.328125 

•34375 

•359375 

•3750 

■390625 

.40925 

.421S75 

■4375 

•453125 

.46875 

.4£4375 

.5000 


515625 

53125 

546875 

5625 

57^125 

5<)375 

609375 

6250 

640625 

65625 

671875 

6875 

703125 

7>875 

734375 

7500 

765625 

78125 

796875 

8125 

828125 

84375 

859375 

8750 

890625 

90625 

921875 

9375 

953125 

96875 

984375 


215 


216  TABLE  II.— COMMON  LOGARITHMS 

Common  Logarithms 


n 

OI2     3456789 

10 

00000 

00432 

00860 

01284 

01703 

02119 

02531 

02938 

03342 

03743 

ii 

04139 

o4532 

04922 

05308 

05690 

06070 

06446 

06819 

0718S 

o7555 

12 

07918 

08279 

08636 

0S991 

09342 

09691 

10037 

10380 

10721 

11059 

13 

"394 

11727 

12057 

123S5 

12710 

13033 

*3354 

13672 

13988 

1430 1 

14 

14613 

14922 

15229 

15534 

15836 

16137 

1643S 

16732 

17026 

173*9 

IS 

17609 

17898 

18184 

18469 

18752 

19033 

19312 

i959o 

19866 

20140 

iG 

20412 

206S3 

20952 

21219 

214S4 

2174S 

22011 

22272 

22531 

22789 

17 

2304S 

23300 

23553 

23805 

24055 

24304 

24551 

24797 

25042 

252S5 

18 

25527 

25768 

26007 

26245 

26482 

26717 

26951 

27184 

27416 

27646 

19 

27S75 

28103 

28330 

28556 

2S7S0 

29003 

29226 

29447 

29667 

298SS 

20 

30103 

303=0 

30535 

30750 

30963 

3**75 

3i327 

3*597 

31806 

3-OI5 

21 

32222 

3242S 

32634 

32S3S 

33041 

33244 

33445 

33646 

33846 

34044 

22 

34242 

34439 

34635 

34S30 

35025 

352i3 

354i  1 

35603 

35793 

35984 

=  3 

36173 

36361 

36549 

36736 

36922 

37107 

37291 

37475 

3765S 

37840 

24 

3S021 

3S202 

3S382 

38561 

38739 

3S917 

39094 

39270 

39445 

39620 

25 

39794 

39967 

40140 

40312 

40483 

40654 

40S24 

40993 

41162 

4*33° 

26 

41497 

41664 

41830 

41996 

42160 

42325 

4248S 

42651 

42813 

42975 

27 

43*36 

43297 

43457 

43616 

43775 

43933 

44091 

4424S 

44404 

44560 

28 

44716 

44S71 

45025 

45179 

45332 

45484 

45637 

45788 

45939 

46090 

29 

46240 

463S9 

4653S 

466S7 

46835 

46982 

47129 

47276 

47422 

47567 

30 

47712 

47857 

48001 

48144 

48287 

4S430 

4S572 

4S714 

48855 

48996 

31 

49*36 

49276 

49415 

49554 

49693 

49S31 

49969 

50106 

50243 

50379 

32 

50515 

50651 

50786 

50920 

51055 

51188 

51322 

5*455 

5*587 

51720 

33 

SlSSi 

51983 

52114 

52244 

52375 

52504 

52634 

52763 

52892 

53020 

34 

53148 

53275 

53403 

53529 

53656 

53782 

53908 

54033 

54158 

54283 

35 

54407 

54531 

54654 

54777 

549oo 

55023 

55145 

55267 

55388 

55509 

36 

55630 

55751 

55871 

5599i 

56110 

56229 

56348 

56467 

56585 

56703 

37 

56S20 

56937 

57054 

57*7* 

57287 

57403 

57519 

57634 

57749 

57864 

38 

57978 

5S092 

58206 

5832o 

58433 

58546 

58659 

5S771 

588S3 

58995 

39 

59106 

59218 

59329 

59439 

5955o 

59660 

5977o 

59879 

59988 

60097 

40 

60206 

60314 

60423 

60531 

60638 

60746 

60853 

6o959 

61066 

61172 

4i 

61278 

61384 

61490 

6iS95 

61700 

61S05 

61909 

62014 

62118 

62221 

42 

62325 

62428 

62531 

62634 

62737 

62S39 

62941 

63043 

63*44 

63246 

43 

63347 

63448 

63548 

63649 

63749 

63849 

63949 

64048 

64147 

64246 

44 

64345 

64444 

64542 

64640 

64738 

64836 

64933 

65031 

65128 

65225 

45 

65321 

65418 

65514 

65610 

65706 

65801 

65896 

65992 

66087 

66181 

46 

66276 

66370 

66464 

66558 

66652 

66745 

66839 

66932 

67025 

67117 

47 

67210 

67302 

67394 

67486 

67578 

67669 

67761 

67852 

67943 

6S034 

48 

68124 

68215 

6S30S 

6S395 

68485 

68574 

6S664 

6S753 

6S842 

68931 

49 

69020 

6910S 

69197 

69285 

59373 

69461 

69548 

69636 

69723 

69S10 

50 

69897 

69984 

70070 

70157 

70243 

70329 

70415 

70501 

705S6 

70672 

51 

70757 

70842 

70927 

71012 

71096 

71181 

71265 

7*349 

7*433 

7*5*7 

52 

71600 

716S4 

71767 

71850 

7*933 

72016 

72099 

72181 

72263 

72346 

53 

72428 

72509 

72S9I 

72673 

72754 

72835 

72916 

72997 

73078 

73*59 

54 

73239 

7332o 

73400 

7348o 

7356o 

73640 

737*9 

73799 

73878 

73957 

01234      56789 

TABLE  II.— COMMON  LOGARITHMS 
of  Numbers  from  000  to  999 


217 


■ 

01      2      34      56789 

55 

74°36 

74i  1 5 

74194 

74273 

7435 1 

74429 

74507 

74586 

74663 

74741 

56 

74819 

74896 

74974 

75051 

75128 

75205 

75282 

75358 

75435 

755** 

57 

75587 

75664 

75740 

758i5 

75S91 

75967 

76042 

76118 

76193 

76268 

58 

76343 

76418 

76492 

76567 

76641 

76716 

76790 

76864 

76938 

77012 

59 

77085 

77IS9 

77232 

77305 

77379 

77452 

77525 

77597 

77670 

77743 

60 

778i5 

77887 

77960 

78032 

78104 

7SI76 

78247 

78319 

78390 

78462 

61 

78533 

78604 

78675 

78746 

78817 

7SSSS 

78958 

79029 

79099 

79169 

62 

792/'9 

79309 

79379 

79449 

795i8 

79588 

79657 

79727 

79796 

79865 

63 

79934 

80003 

£0072 

80140 

80209 

80277 

80346 

80414 

804S2 

80550 

64 

80618 

806S6 

80754 

80821 

80889 

80956 

81023 

81090 

81158 

81224 

65 

81291 

81358 

81425 

81491 

81558 

81624 

81690 

8*757 

81823 

81889 

66 

8i954 

82020 

82086 

82151 

82217 

82282 

82347 

82413 

82478 

82543 

67 

82607 

82672 

82737 

82802 

82866 

82930 

82995 

83059 

83*23 

83187 

68 

83251 

83315 

83378 

83442 

83506 

83569 

83632 

83696 

83759 

83822 

69 

83885 

83948 

840 1 1 

84073 

84136 

84198 

84261 

84323 

84386 

84448 

70 

84510 

84572 

84634 

84696 

84757 

84819 

84880 

84942 

85003 

85065 

71 

85126 

85187 

85248 

85309 

8537o 

85431 

85491 

85552 

85612 

85673 

72 

85733 

85794 

S5854 

85914 

85974 

86034 

86094 

86153 

£6213 

86273 

73 

86332 

863:12 

86451 

S6510 

S6570 

86629 

86688 

86747 

86806 

86864 

74 

86923 

86982 

87040 

87099 

87157 

87216 

87274 

87332 

87390 

87448 

75 

87506 

87564 

87622 

87679 

87737 

87795 

87852 

87910 

87967 

88024 

76 

8S0S1 

S8138 

S8195 

88252 

88309 

88366 

88423 

S8480 

S8536 

88593 

77 

88649 

8S705 

88762 

S8818 

88874 

88930 

88986 

89042 

89098 

89*54 

78 

89209 

89265 

89321 

89376 

89432 

89487 

89542 

89597 

89653 

89708 

79 

89763 

89818 

89873 

89927 

89982 

90037 

90091 

90146 

90200 

90255 

80 

90309 

90363 

90417 

90472 

90526 

90580 

90634 

90687 

90741 

90795 

81 

90849 

90902 

90956 

91009 

91062 

91116 

91169 

91222 

9*275 

91328 

82 

91381 

9*434 

91487 

9*54© 

9*593 

9*645 

91698 

9*75* 

91803 

91855 

83 

91908 

91960 

92012 

92065 

92117 

92169 

92221 

92273 

92324 

92376 

84 

92428 

924S0 

92531 

925S3 

92634 

92686 

92737 

92788 

92840 

92891 

85 

92942 

92993 

93044 

93095 

93146 

93*97 

93247 

93298 

93349 

93399 

86 

9345° 

935oo 

93551 

93601 

93651 

93702 

93752 

93S02 

93852 

93902 

87 

93952 

94002 

94052 

94101 

94iSi 

94201 

94250 

94300 

94349 

94399 

88 

94448 

94498 

94547 

94596 

94645 

94694 

94743 

94792 

94841 

94890 

89 

94939 

94988 

95036 

95oSs 

95*34 

95182 

95231 

95279 

95328 

95376 

90 

95424 

95472 

95521 

95569 

956i7 

95665 

95713 

9576i 

95809 

95856 

91 

95904 

95952 

95999 

96047 

96095 

96142 

96190 

96237 

962S4 

96332 

92 

96379 

96426 

96473 

96520 

96567 

96614 

96661 

96708 

96755 

96802 

93 

96848 

96895 

96942 

969S8 

97035 

97081 

97128 

97*74 

97220 

97267 

94 

973*3 

97359 

97405 

974S1 

97497 

97543 

97589 

97635 

97681 

97727 

95 

97772 

97818 

97864 

97909 

97955 

98000 

9S046 

98091 

98137 

98182 

96 

98227 

98272 

98318 

98363 

98408 

98453 

98498 

9S543 

98588 

98632 

97 

9S677 

98722 

98767 

98811 

98856 

98900 

9894S 

98989 

99034 

99078 

98 

99123 

99167 

99211 

99255 

99300 

99344 

99388 

99432 

99476 

99520 

99 

99564 

99607 

99651 

99695 

99739 

99782 

99826 

99870 

999*3 

99957 

01-3     4     56     789 

218 


.TABLE  III.— LOGARITHMS  OP 

Common  Logarithms 


JLOG  TANOF.NT 


Angle 


30 


40 


50 


6o' 


13 
14 

I5C 

16 
17 

18 
i9 


23 
24 

2SC 

26 

27 

28 

29 

30° 

31 

32 

33 
34 

35° 

36 

37 

38 
39 

40' 

4i 

42 

43 

44 


—  00 
2.24192 
2.54308 
2.71940 
2.S4464 

2.94195 
T. 02162 
T. 08914 
1. 14780 
T.19971 

T. 24632 
T.28S65 
i-32747 
i-36336 
1.39677 

T. 42805 
i-4575° 
I-48534 
1.51178 

I-53697 
T.  56107 
1. 58418 
1. 60641 

1-62785 
1.64858 

T. 66867 
T. 68818 
T. 70717 
T. 72567 
1-74375 
1. 76144 
1.77877 

f- 79579 
1. 81252 
I.82S99 

i.84523 
1. 86126 
1.87711 
T.  89281 
T.  90837 

1. 92381 
T. 93916 
1-95444 
1.96966 


3-46373 
2.30888 
2.577S8 
2.74292 
2.S6243 

2.95627 
1. 03361 
1.09947 
1. 15688 
T. 20782 

1-25365 
1-29535 
I-33365 
1.36909 

1. 40212 

1.4330S 
1.46224 
T.489S4 
T. 51606 
I. 54106 

T. 56498 
I.58794 
1. 61004 

1-63135 
1.65197 

1. 67196 
1. 69138 
r. 71028 
1. 72872 
f • 74673 

t-  76435 
f.  78163 
1.  79860 
f.81528 
1.83171 

T.  84791 
1.86392 
1.87974 
1. 89541 

i^ogs 

i.  92638 

F.94171 
f.  95698 
1 .97219 
1.98737 


3.76476 
2 .36689 
2 . 61009 
2.76525 
2-87953 
2.970:; 

1.045  8 
T. 10956 
f- 16577 
1. 21578 

T. 26086 
i-3OI95 
1-33974 
1-37476 
T. 40742 

T. 43806 
T. 46694 
1.49430 
T. 52031 
I.54512 
1.56887 
1. 59168 
1. 61364 
1.63484 
1-65535 
1.67524 
1.69457 
i-7I339 
L73I75 
1.74969 

T. 76725 
T.7844S 
I . 80140 
1. 81803 
I.83442 

1-85059 
1.86656 
1.88236 
1. 89801 
I-9I353 


3.94086 
2.41807 
2. 64009 
2. 78649 


2.06581 
2.46385 
2.66816 
2 . S0674 
2.91185 

98358  2.99662 


.92894 
.94426 
•95952 
.97472 
.98989 


i".  05666 
1.11943 
1. 17450 
T. 22361 

T. 26^97 
1.30846 

1-3457'' 
^•38035 
I. 41266 

1.44299 
1. 47160 
1.49872 
i-52452 
I.549I5 

1.57274 
I-59540 
1. 61722 
1.63830 
1.65870 

1.67850 
1.69774 
1. 71648 
1.73476 
1.75264 

1. 77015 
1.78732 
1. 80419 
1.82078 
1-83713 
1-85327 


1.06775 
1. 12909 
T. 18306 
I. 23130 

T. 27496 
I. 31489 
i-35!7o 
I-385S9 
T. 41 784 

1.44787 
1 .47622 

i-5°3" 
1.52870 

I.553I5 
1.57658 
I.59909 
1.62079 

1. 64175 
1.66204 

T. 68174 
r . 700S9 
f-71955 
1-73777 

1-75558 


2. 16273 
2.50527 
2.69453 
2.82610 
2.92716 

1 .00930 
T. 07858 
1-13854 
1 . 19146 
1.23887 

T. 28186 
T. 32122 
f-35757 
£•39136 
1.42297, 

I. 45271  1.45750 
r .48080  1 . 48534 


2. 24192 
2.54308 
2.71940 
2 . 84464 
2.94195 

T. 02162 
T.0S914 
I. 14780 
T.19971 
1.24632 

r. 28065 
3-32747 
1-36336 
1.39677 
1.42S05 


1. 90061 
T.91610 

i-93I5° 
T. 94681 
r . 96205 
1.97725 
1.99242 


I.77303 
1. 79015 
1.80697 
T. 82352 
1.83984 

1.85594 
T. 87185 
1.88759 
1.90320 
I. 91868 

1.93406 
1-94935 
I-96459 
1.97978 

1-99495 


1.50746 
1-53285 
1-557" 

i. 58039 
1.60276 

3-62433 
1-64517 
1-66537 

1.68497 
1 . 70404 
c . 72262 
c. 74077 
c- 75852 

i- 77591 
i.  79297 

i.  80975 
".82626 


1.51178 

f-53697 
1. 56107 

1. 58418 
1 . 60641 
T. 62785 
T. 64858 
T. 66867 

i. 68818 

i. 70717 
1.72567 

1-74375 
1. 76144 

f.  77877 

t- 79579 
t. 81252 
3.82899 

1. 84254 I 1.84523 

1.85860  1. 86126 
1.87448  T.87711 


1. 89020 
3.90578 
1.92125 

1. 9366i 

i-95'9° 
1. 96712 
1. 98231 
1.99747 


1.89281 
T. 90837 
1. 92381 

1.93916 
1-95444 
1 .96966 
T. 98484 
0.00000 


89 

88 
87 

86 
8S 
84 
83 

82 
81 
8oe 

79 
78 
77 
76 
75° 

74 
73 
72 
7i 
7o° 

69 
68 
67 
66 
65° 

64 
63 
62 
61 
6o° 

59 

58 
57 
56 
55  ; 

54 
53 

5  2 
51 

goe 

49 

48 
47 
46 
45° 


60' 


So' 


40 


30' 


Angle 


L.OU  COXA^GliJSX 


TRIGONOMETRIC  FUNCTIONS 
of  Sines  and  Cosines 

LOG  SINE 


219 


Angl 

e   o' 

10'     20'     30'     40' 

50'     60' 

45° 

I.84949 

1.85074 

1. 85200 

1.85324  1.85448 

I. 85571 

1-85693 

44 

46 

1-85693 

T. 85815 

1-85936 

T. 86056 

T. 86176 

T. 86295 

1.S6413 

43 

47 

1. 86413 

1.86530 

T. 86647 

T. 86763 

1.86879 

1.86993 

1. 87107 

42 

48 

f .87107 

T. 87221 

J- 87334 

1.87446 

1-87557 

T. 87668 

1.87778 

41 

49 

1.S777S 

1.87887 

1.87996 

I. 88105 

1. 88212 

T. 88319 

1.88425 

40° 

5o° 

188425 

T. 88531 

1.88636 

T. 88741 

T. 88844 

T. 88948 

1.89050 

39 

Si 

T. 89050 

T. 89152 

1.89254 

I.89354 

i-s9455 

I.89554 

1-89653 

38 

52 

1-89653 

1.89752 

1. 89S49 

1.89947 

1 . 90043 

1. 90139 

1.90235 

37 

53 

1.90235 

i-9°33° 

T. 90424 

1.90518 

1 .90611 

1.90704 

1.90796 

36 

54 

1.90796 

I.90887 

T. 90978 

I. 91069 

1.91158 

1. 9 1 248 

i-9x336 

35° 

55° 

i-9J336 

1-91425 

T.91512 

I-9I599 

T. 91686 

T. 91772 

1-9^857 

34 

56 

1-91857 

1. 91942 

1.92027 

1 .92111 

T. 92194 

1 .92277 

I.92359 

33 

57 

i-923S9 

T. 92441 

1 . 92522 

1 . 92603 

1.92683 

L92763 

1 .92842 

32 

58 

T. 92842 

T. 92921 

I-92999 

1.93077 

I-93I54 

1.93230 

i-933°7 

31 

59 

i-933°7 

1.93382 

1-93457 

!• 93532 

1 . 93606 

1.93680 

1-93753 

30° 

6o° 

1 -93753 

1.93826 

T- 93898 

T. 93970 

1. 94041 

1.94112 

T. 94182 

29 

61 

1. 94182 

1.94252 

1. 94321 

1.94390 

T. 94458 

1.94526 

1-94593 

28 

62 

1-94593 

1 .94660 

1.94727 

1-94793 

T. 94858 

1.94923 

1 .94988 

27 

63 

T.949S8 

i-95°52 

T.95116 

I-95I79 

1-95242 

I- 95304 

I-95366 

26 

64 

1-95366 

I-95427 

1.95488 

1-95549 

1.95609 

1.95668 

1.95728 

25° 

65° 

i-95728 

1.95786 

T. 95844 

T. 95902 

1.95960 

T. 96017 

T. 96073 

24 

66 

T. 96073 

1. 96129 

T. 96185 

1 .96240 

1 .96294 

1.96349 

1 . 96403 

23 

67 

1.96403 

T. 96456 

T. 96509 

1.96562 

T. 96614 

1 .96665 

1 .96717 

22 

68 

1. 96717 

1.96767 

T.9681C 

T.96S6S 

T. 96917 

T. 96966 

T. 97015 

21 

69 

1. 97015 

T. 97063 

1.97111 

1-97*59 

1.97206 

1.97252 

1.97299 

20J 

7o° 

1.97299 

1-97344 

1 -9739° 

1-97435 

T. 97479 

T.97523 

T. 97567 

19 

7i 

I-97567 

1 . 97610 

I-97653 

1 .97696 

I-97738 

1.97779 

1. 97821 

18 

72 

1. 97821 

T. 97861 

1.97902 

T. 97942 

1 . 97982 

1 .98021 

1. 98060 

17 

73 

T.9S060 

T.9S09S 

1.98136 

1. 98174 

T. 982 1 1 

1.98248 

1.98284 

16 

74 

1.98284 

T. 98320 

1-98356 

i.9839i 

1.9S426 

T. 98460 

1.98494 

15° 

75° 

1.98494 

T. 98528 

T. 98561 

I.98594 

1.98627 

T. 98659 

1.98690 

14 

76 

1.98690 

T. 98722 

I-98753 

1.98783 

1-98813 

J-y  ';4.s 

T. 98872 

13 

77 

I.98S72 

1.98901 

1.98930 

1.98958 

1 .9S9S6 

1. 99013 

T. 99040 

12 

78 

1.99040 

T. 99067 

I-99093 

1.99119 

I-99M5 

1. 99170 

i-99J95 

11 

79 

1.99195 

T. 99219 

1.99243 

1.99267 

1 .99290 

1-99313 

1-99335 

IO° 

8o° 

^•99335 

1-99357 

1-99379 

1.99400 

T. 99421 

T. 99442 

T. 99462 

9 

81 

1.99462 

1.99482 

1. 99501 

T. 99520 

I-99539 

1-99557 

1-99575 

8 

82 

1-99575 

1-9959.1 

1 .99610 

1.99627 

T. 99643 

T. 99659 

T. 99675 

7 

83 

I-99675 

T. 99690 

1.99705 

T. 99720 

1-99734 

1.99748 

1. 99761 

6 

84 

T. 99761 

1-99775 

I.99787 

1.99S00 

T. 99812 

T. 99823 

T. 99834 

5° 

85° 

T. 99834 

T. 99845 

T. 99856 

T. 99866 

T. 99876 

T. 99885 

T. 99894 

4 

86 

T. 99894 

T. 99903 

1.99911 

T. 99919 

T. 99926 

1-99934 

1.99940 

3 

87 

1.999+0 

1.99947 

1-99953 

1-99959 

T. 99964 

T. 99969 

1.99974 

2 

88 

1.99974 

T. 99978 

1.99982 

1.99985 

T. 99988 

T. 99991 

1-99993 

1 

89 

1-99993 

1-99995 

1-99997 

1.99998 

1-99999 

0.00000 

0 . 00000 

0° 

60' 

50'     40'      30'     20' 

10'      0'  I 

^ngle 

JUOO  C05.J.NJS 


220 


TABLE  III.— LOGARITHMS  OF 

Common  Logarithms 


tOG  SINE 

Angle   o'     10'     20'     3°'    40'     50'     60' 

0° 

.  —  °o 

3-46373 

3.7647S 

2 ■ 94°s4 

2.06578 

2. 16268 

2. 24186 

§9 

1 

2.24186 

2.30879 

2.3667S 

2.41792 

2.46366 

2.50504 

2.54282 

88 

2 

2.54282 

2.57757 

'2.60973 

2 .63968 

2.66769 

2 . 69400 

2. 71880 

87 

3 

3. 71880 

2 . 74226 

2.76451 

2.7S568 

2.80585 

2.S2513 

2.84358 

86 

4 

2.84358 

2.86128 

2.S7S29 

2.89464 

2 .91040 

2.92561 

2 . 94030 

85° 

5° 

2.94030 

2.95450 

2.96825 

2.93157 

2.99450 

T. 00704 

T. 01923 

84 

6 

1. 01923 

1. 03109 

T. 04262 

1.05386 

1. 0648 1 

1.07548 

T. 08589 

83 

7 

1.08589 

T. 09606 

T. 10599 

1.11570 

T.12519 

I-I3447 

1. 14356 

82 

8 

I.I4356 

1.  I5-M5 

1. 16116 

T. 16970 

1.17!  37 

T.1S628 

I-I9433 

81 

9 

T- 19433 

1.20223 

1.20999 

1. 21761 

1.22509 

1.23244 

1.23967 

8o° 

IO° 

T.  23967 

1.24677 

T. 25376 

1 . 26063 

T. 26739 

T. 27405 

T. 28060 

79 

11 

1. 28060 

T. 28705 

1.29340 

T. 29966 

1.30582 

1:31189 

1. 31788 

78 

12 

1.317SS 

i-32378 

1.32960 

1-33534 

1. 34100 

1.3465S. 

1.35209 

77 

13 

T.  35209 

T. 35752 

T. 36289 

T. 36819 

1 -37341 

I-37858 

1.38368 

76 

14 

1.38368 

I.38871 

I.39369 

T. 39860 

1.40346 

1.40S25 

1. 41300 

75° 

15° 

1. 41300 

1. 41768 

T. 42232 

T. 42690 

i-4.'i43 

I-4359I 

1.44034 

74 

16 

1.44034 

T. 44472 

T. 44905 

7.45334 

I.45758 

I.4617S 

1.46594 

73 

17 

T. 46594 

1.47005 

T.47411 

T. 47814 

T. 48213 

1.48607 

1.48998 

72 

18 

T. 48998 

i-493s5 

1.49768 

T. 50148 

1-50523 

I.50S96 

T. 51264 

7i 

19 

T. 51264 

I. 51629 

i-5I99i 

i-5235o 

1-52705 

I-53056 

1 • 53405 

70° 

20° 

I-53405 

I.5375I 

I.54093 

1-54433 

T. 54769 

i-55102 

1-55433 

69 

21 

1-55433 

i.5576i 

1.56085 

1 . 56408 

1.56727 

1.57044 

1-57358 

68 

22 

7-5735S 

1.57669 

I-57978 

1.58284 

1.58588 

T.5S889 

T.5918S 

67 

23 

T. 59188 

1.59484 

1-59778 

1.60070 

1.60359 

i . 60646 

1. 60931 

66 

24 

I. 60931 

1.61214 

T. 61494 

T. 61773 

T. 62049 

1.62323 

1-62595 

65° 

25° 

^.62595 

T. 62865 

1-63133 

1.63398 

T. 63662 

1.63924 

T. 64184 

64 

26 

1. 64184 

T. 64442 

T. 64698 

1.64953 

1.65205 

1-65456 

I-65705 

63 

27 

i.65705 

1.65952 

T. 66197 

1. 66441 

T.666S2 

1.66922 

1.67161 

62 

28 

T.67161 

1-67398 

1-67633 

T.67S66 

I.6S098 

T. 68328 

1-68557 

61 

29 

T.6S557 

T. 68784 

1. 69010 

1.69234 

1.69456 

1.69677 

1.69897 

60  ° 

3o° 

T. 69897 

T.70115 

T. 70332 

I.70547 

T. 70761 

T. 70973 

1.71184 

59 

3i 

T.71184 

i-7I393 

1. 71602 

1. 71809 

T. 72014 

T. 72218 

1. 72421 

58 

32 

1. 72421 

T. 72622 

1.72823 

T. 73022 

T. 73219 

T. 73416 

1.73611 

S7 

33 

1.73611 

1.73805 

1-73997 

I. 74189 

1-74379 

1 . 74568 

L74756 

56 

34 

1.74756 

1-74943 

1.75128 

I.753I3 

1.75496 

1.75678 

1.75859 

55° 

35° 

1.75859 

1.70039 

T. 76218 

1.76395 

T. 76572 

T. 76747 

I.76922 

54 

36 

1. 76922 

T. 77095 

1.77268 

1.77439 

1.77609 

1.7777S 

1.77946 

53 

37 

1.77946 

T.78113 

1.7S2S0 

1.78445 

1 . 78609 

1.78772 

I.78934 

52 

38 

1.78934 

T. 79095 

1.79256 

I-794I5 

1-79573 

I-7973I 

1.79887 

5i 

39 

1.79887 

1 . 80043 

T. 80197 

1. 80351 

1.80504 

T. 80656 

1.80807 

50° 

400 

T. 80807 

T. 80957 

T.81106 

T. 81254 

I. 81402 

T. 81549 

T. 81694 

49 

41 

1 .81694 

1.81839 

T. 81983 

T. 82126 

1.82269 

1. 82410 

1. 82551 

48 

42 

1. 82551 

1. 82691 

T. 82830 

T. 82968 

1. S3 106 

1.83242 

1.83378 

47 

43 

1.83378 

i-83513 

T. 83648 

T.837S1 

7. 83914 

1 . 84046 

T. 84177 

46 

44 

1. 84177 

1.84308 

1.84437 

1.84566 

T. 84694 

I.84822 

1 . 84949 

45° 

60'     50'     4c'     30'     20'     io'     o'  . 

\ngle 

u 

)G  COS! 

NE 

TRIGONOMETRIC  FUNCTIONS 
ol  Tangents  and  Cotangents 

EOG  TANGENT 


221 


Angl 

S        <>'                  IO'                   20'                30'                  40'                 50'                 60' 

45° 

O.OOOOO 

0.00253 

0.00505 

0.00758 

O.OIOII 

0.01263 

0.01516 

44 

46 

O.OI516 

0.01769 

0.02022 

0.02275 

0.02528 

0.02781 

0.03034 

43 

47 

O.O3O34 

0.03288 

0.03541 

0.03795 

0.0404S 

0.04302 

0.04556 

42 

48 

0.04556 

0.04810 

0.05065 

0.05319 

0.05574 

0.05S29 

c. 06084 

41 

49 

0.06084 

0.06339 

0.06594 

0.06S50 

0.07106 

0.07362 

0.07619 

40° 

5o° 

0.07619 

0.07875 

0.08132 

0.0S390 

0.08647 

0.08905 

0.09163 

39 

5i 

0.09163 

0.09422 

0.096S0 

0.09939 

0. 10199 

0.10459 

0. 10719 

38 

52 

0. 10719 

0. I09S0 

0. 11241 

0. 11502 

0. 11764 

0. 12026 

0. 12289 

37 

53 

0. 12289 

0.12552 

0. 12815 

0.13079 

0.13344 

0. 13608 

0.13874 

36 

54 

0.13874 

0. 14140 

0. 14406 

0.14673 

0. 14941 

0. 15209 

O.I5477 

35° 

55° 

O.I5477 

0.15746 

0. 16016 

0. 16287 

0.16558 

0. 16829 

0. 17101 

34 

56 

0. 17101 

0.17374 

0. 1764S 

0. 17922 

0. 18197 

0. 18472 

0.18748 

33 

57 

0.18748 

0. 19025 

0.19303 

0.195S1 

0. 19860 

0. 20140 

0. 20421 

32 

58 

0. 20421 

0. 20703 

2.20985 

0. 21268 

0.21552 

0.21837 

0. 22123 

31 

59 

0.22123 

0.22409 

0.22697 

0.22985 

0.23275 

0.23565 

0.23856 

30° 

60  ° 

0.23856 

0. 24148 

0.24442 

0.24736 

0.25031 

0.25327 

0.25625 

29 

61 

0.25625 

0.25923 

0. 26223 

0. 26524 

0. 26825 

0. 27128 

0.27433 

28 

62 

0.27433 

0.27738 

0.28045 

0.28352 

0. 28661 

0.28972 

0.29283 

27 

63 

0. 292S3 

0.29596 

0.29911 

0.30226 

0.30543 

0.30S62 

0.31182 

26 

64 

0.31182 

o.3IS03 

0.31826 

0.32150 

0.32476 

0.32804 

o.33i33 

25° 

65° 

o.33i33 

0.33463 

0.33796 

0.34130 

0.34465 

0.34803 

o.35M2 

24 

66 

0.35142 

0.35483 

0.35825 

0.36170 

0.36516 

0.36865 

0.37215 

23 

67 

0.37215 

0.37567 

0.37921 

0.3827S 

0.38636 

0.38996 

o.39359 

22 

68 

o.39359 

0.39724 

0.40091 

0.40460 

0.40832 

0.41206 

0.41582 

21 

69 

0.41582 

0.41961 

0.42342 

O.42726 

o.43rl3 

0.43502 

0.43893 

20° 

7o° 

0.43893 

0.44288 

0.44685 

0.45085 

0.45488 

0.45894 

0.46303 

19 

7i 

0.46303 

0.46715 

0.47130 

0.4754S 

0.47969 

0.48394 

0.48822 

l8 

72 

0.48822 

0.49254 

0.49689 

0. 50128 

0.50570 

0. 51016 

0.51466 

17 

73 

0. 51466 

0. 51920 

0.52378 

0.52840 

0.53306 

o.53776 

0.54250 

16 

74 

0.54250 

0.54729 

0.55213 

0.5S70I 

0.56194 

0.56692 

o.57i95 

15° 

75° 

o.57i95 

0.57703 

0. 58216 

0.58734 

0.59258 

o.59788 

0.60323 

14 

76 

0.60323 

0.60864 

0. 61411 

0.61965 

0.62524 

0.63091 

0.63664 

13 

77 

0. 63664 

0.64243 

0.64830 

0.65424 

0.66026 

0.66635 

0.67253 

12 

78 

0.67253 

0.67878 

0.68511 

0.69154 

0.69805 

0.70465 

o.7"35 

II 

79 

0.7H3S 

0. 71814 

0.72504 

0.73203 

°-739I4 

0.74635 

0.75368 

10° 

8o° 

o.75368 

0.76113 

0.76870 

0.77639 

0. 78422 

0. 79218 

0. 80029 

9 

81 

0.80029 

0.80854 

0. 81694 

0.82550 

0.83423 

0.84312 

0.85220 

8 

82 

0. 85220 

0.86146 

0. 87091 

0.88057 

0.89044 

0.90053 

0.91086 

7 

83 

0. 91086 

0.92142 

0.93225 

o.94334 

0.95472 

0.96639 

0.97838 

6 

84 

0.97838 

0.99070 

1.00338 

1 .01642 

1.02987 

1.04373 

1.05805 

5° 

85° 

1.05805 

1.07284 

1. 08815 

1. 10402 

1. 12047 

1. 13757 

I.I5S36 

4 

86 

I-I5536 

1. 17390 

1. 19326 

1.21351 

1-23475 

1.25708 

1. 28060 

3 

87 

1. 28060 

1.30547 

i.33l84 

I-35991 

1. 38991 

1. 42212 

1.45692 

2 

88 

1.45692 

1.49473 

I.536I5 

1. 58193 

1.63311 

1.69112 

1.75808 

I 
0 

89 

1.75808 

1.83727 

1. 93419 

2.05914 

2.23524 

2.53627 

00 

1      ° 

6o'            50'            40'           30'            20'             10'             0' 

4ngle 

LOG 

COTAN 

GENT 

222 


TABLE  IV.— TRIGONOMETRIC  FUNCTIONS 

Natural  Sines 


SINE 


Angle 

0'      10'      20'     30'      4o'     So'      60' 

o° 

0.00000 

0.00291 

0.00582  0.00873 

0.01164 

0.01454  0.01745 

89 

z 

O.01745 

0.02036 

0.02327  0.02618 

0.02908 

0.03199  0.03490 

88 

2 

O.03490 

0.03781 

0.04071 

0.04362 

0.04653 

0.04943  0.05234 

87 

3 

O.05234 

0.05524 

0.05814 

0.06105 

0.06395 

0.06685 

0.06976 

86 

4 

O.06976 

0.07266 

0.07556 

0.0784c 

0.08136 

0.08426 

0.08716 

850 

5° 

0.08716 

0.09005 

0.09295 

0.09^85 

0.09874 

0.10164 

0.10453 

84 

6 

O.IO453 

0.10742 

0.11031 

0. 11320 

0.1 1609 

0.11898 

0.12187 

83 

7 

0.12187 

0.12476 

0.12764 

0.13053 

O.I334I 

0.13629 

0.13917 

82 

8 

O.I391? 

0.14205 

O.I4493 

0. 14781 

0.15069 

O.I5356 

0.15643 

81 

9 

0.15643 

0.15931 

0. 16218 

0.16505 

0.16792 

0.17078 

O.I7365 

8o° 

IO° 

0.1736S 

0.17651 

o.i7937 

0. 18224 

0.18509 

0.18795 

0.19081 

79 

ii 

0. 19081 

0. 19366 

0.19652 

O.I9937 

0. 20222 

0. 20507 

0.20791 

78 

12 

0. 20791 

0.21076 

0. 21360 

0.21644 

0. 21928 

0.22212 

0.22495 

77 

13 

0.22495 

0.22778 

0. 23062 

0.23345 

0.23627 

0. 23910 

0.24192 

76 

14 

0.24192 

0.24474 

0.24756 

0.25038 

0.25320 

0.25601 

0.25882 

75° 

15° 

0.25882 

0.26163 

0.26443 

0.26724 

0. 27004 

0. 27284 

0.27564 

74 

16 

0.27564 

0.27843 

0.28123 

0. 28402 

0.28680 

0.28959 

0.29237 

73 

17 

0.29237 

0.29515 

0.29793 

0.30071 

0.30348 

0.30625 

0.30902 

72 

18 

0.30902 

o.3"78 

0.31454 

0.31730 

0.32006 

0.32282 

0.32557 

71 

i9 

o.32557 

0.32832 

0.33106 

0.33381 

0.33655 

0.33929 

0.34202 

70° 

20° 

0.34202 

o.34475 

0.34748 

0.35021 

0.35293 

0.35565 

0.35837 

69 

21 

o.35837 

0.36108 

0.36379 

0.36650 

0.36921 

0.37191 

0.37461 

68 

22 

0.37461 

0.37730 

o.37999 

0.38268 

0.38537 

0.38805 

0.39073 

67 

23 

o.39073 

0.39341 

0.3960S 

0.39875 

0.40142 

0.40408 

0.40674 

66 

24 

0.40674 

0.40939 

0.41204 

0.414C9 

0.41734 

0.41998 

0.42262 

65° 

25° 

0.42262 

0.42525 

0.42788 

0.43051 

0.43313 

0.43575 

0.43837 

64 

26 

0.43837 

0. 44098 

o.44359 

0.44620 

0.448S0 

0.45140 

0.45399 

63 

27 

o.45399 

0.45658 

0.45917 

0.46175 

0.46433 

0.46690 

0.46947 

62 

28 

0.46947 

0.47204 

0.47460 

0.47716 

0.47971 

0.48226 

0.48481 

61 

29 

0.48481 

o.48735 

0.48989 

0.49242 

0.49495 

0.49748 

0.50000 

6o° 

30° 

0. 50000 

0.50252 

0.50503 

0.50754 

0. 51004 

0.51254 

O.51504 

59 

31 

0.51504 

o.5i753 

0. 52002 

0.52250 

0.52498 

0.52745 

0.52992 

58 

32 

0.52992 

0.53238 

0.53484 

0.5373° 

0.53975 

0.54220 

0.54464 

57 

33 

0.54464 

0.54708 

0.54951 

o.55T94 

o.55436 

0.55678 

o.559i9 

56 

34 

0.55919 

0.56160 

0.56401 

0.56641 

0.56880 

0.57119 

o.57358 

55° 

35° 

0.57358 

o.57596 

0.57833 

0.58070 

0.58307 

0.58543 

0.58779 

54 

36 

0.58779 

0. 59014 

0.59248 

0.59482 

0.59716 

o.59949 

0.60182 

53 

37 

0.60182 

0.60414 

0.60645 

0.60876 

0.61107 

0.61337 

0.61566 

52 

38 

0.61566 

0.61795 

0.62024 

0.62251 

0.62479 

0.62706 

0.62932 

51 

39 

0.62932 

0.63158 

0.63383 

0.63608 

0.63832 

0.64056 

0.64279 

50° 

40° 

0.64279 

0.64501 

0.64723 

0.64945 

0.65166 

0.65386 

0.65606 

49 

41 

0. 65606 

0.65825 

0.66044 

0.66262 

0.66480 

0.66697 

0.66913 

48 

42 

0.66913 

0.67129 

0.67344 

0.67559 

0.67773 

0.67987 

0.68200 

47 

43 

0. 68200 

0.68412 

0.68624 

0.68835 

0.69046 

0.69256 

0. 69466 

46 

44 

0.69466 

0.69675 

0.69883 

0. 70091 

0. 70298 

0.70505 

0. 7071 1 

45 

60'    50'     40'    30'     20'    10'     0'  / 

Lngle 

CO&UN1 

TABLE  IV.— TRIGONOMETRIC   FUNCTIONS 
and  Cosines 


223 


SINE 

Angl 

e    o'    io'      20'    30'     40'      5o'    60' 

44 

45° 

0.70711 

0. 70916 

0. 71121 

0.71325 

0.71529 

0.71732 

o.7J934 

46 

0-71934 

0. 72136 

o.72337 

0.72537 

0.72737 

0.72937 

o.73i35 

43 

47 

0-73135 

o.73333 

o.7353i 

0.73728 

0.73924 

0. 74120 

0.74314 

42 

48 

0.743M 

0.74509 

0.74703 

0. 74896 

0.75088 

0. 75280 

o.7547i 

4i 

49 

0.7S47I 

0.75661 

0-75851 

0. 76041 

0. 76229 

0. 76417 

0. 76604 

40° 

go0 

0. 76604 

0. 76791 

0.76977 

0.77162 

0-77347 

0.77S3I 

0.77715 

39 

51 

0.77715 

0.77897 

0.78070 

0.78261 

0. 78442 

0.78622 

0.78801 

38 

53 

0.7SS01 

0.78980 

0.7915C 

0.79335 

0.79512 

0. 7968S 

0. 79S64 

37 

53 

0.79864 

0.80038 

0.80212 

0.80386 

0.80558 

0.80730 

0. 80902 

36 

54 

0.80902 

0.81072 

0.81242 

0.81412 

0.81580 

0.81748 

0.81915 

35° 

55° 

0.81915 

0.82082 

0.82248 

0.82413 

0.82577 

0.82741 

0.82904 

34 

56 

0.82904 

0.83066 

0.83228 

0.83389 

0.83549 

0.83708 

0.83867 

33 

37 

0.83867 

0.84025 

0.84182 

0.84339 

0.84495 

0.84650 

0.84805 

32 

58 

0.84805 

0.84959 

0.85112 

0.85264 

0.85416 

0.85567 

0.85717 

31 

59 

0.85717 

0.85866 

0.86015 

0.86163 

0.S6310 

0.86457 

0.86603 

30° 

60  ° 

0.86603 

0.86748 

0.86892 

0.87036 

0.87178 

0.87321 

0.87462 

29 

61 

0.87462 

0.87603 

0.87743 

0.878S2 

0.88020 

0.88158 

0.88295 

28 

62 

0.88295 

0.88431 

0.88566 

0.88701 

0.88835 

0.88968 

0. 89101 

27 

63 

0.89101 

0.89232 

0.89363 

0.89493 

0.89623 

0.89752 

0.89879 

26 

64 

0.89879 

0.90007 

0.90133 

0.90259 

0.90383 

0.90507 

0.90631 

25° 

65° 

0.90631 

0.90753 

0.90875 

0.90996 

0.91116 

0.91236 

0.91355 

24 

66 

0.91355 

0.91472 

0.91590 

0. 91706 

0.91822 

0.91936 

0.92050 

23 

67 

0. 92050 

0.92164 

0.92276 

0.92388 

0.92499 

0.92609 

0.92718 

22 

68 

0.92718 

0.92827 

0.92935 

0.93042 

0.93148 

0.93253 

0-93358 

21 

69 

0.93358 

0.93462 

0.93565 

0.93667 

0.93769 

0.93869 

0.93969 

20° 

70° 

0.93969 

0.94068 

0.94167 

0.94264 

0.94361 

0.94457 

0-94552 

19 

7i 

0.94552 

0.94646 

0.94740 

0.94832 

0.94924 

0.95015 

0.95106 

18 

72 

0.95106 

0.95195 

0.95284 

0.95372 

0.95459 

0.95545 

0.95630 

17 

73 

0.95630 

0.95715 

0-95799 

0.95882 

0.95964 

0. 96046 

0. 96126 

16 

74 

0.96126 

0.96206 

0.96285 

0.96363 

0.96440 

0.96517 

0.96593 

15° 

75° 

0.96593 

0.96667 

0.96742 

0.96815 

0.96887 

0.96959 

0.97030 

14 

76 

0.97030 

0.97100 

0.97169 

0.97237 

0.97304 

0.97371 

o.97437 

13 

77 

o-97437 

0.97502 

0.97566 

0.97630 

0.97692 

0.97754 

0.97815 

12 

78 

0.97815 

0.97875 

0.97934 

0.97992 

0.9S050 

0.98107 

0.98163 

11 

79 

0.98163 

0.98218 

0.98272 

0.98325 

0.98378 

0.98430 

0.98481 

10° 

8o° 

0.98481 

0-98531 

0.98580 

0.98629 

0.98676 

0.98723 

0.98769 

9 

81 

0.98769 

0.98814 

0.9885S 

0.98902 

0.98944 

0.98986 

0.99027 

8 

82 

0.99027 

0.99067 

0.' 99 106 

0.99144 

0. 99182 

0.99219 

0.99255 

7 

83 

0.99255 

0.99290 

0.99324 

0.99357 

0.99390 

0.99421 

0.99452 

6 

84 

0.99452 

0.99482 

0.99511 

0.99540 

0.99567 

0-99594 

0.99619 

5° 

850 

0. 99619 

0.99644 

0.99668 

0.99692 

0.99714 

0.99736 

0.99756 

4 

86 

0.99756 

0.99776 

0.99795 

0.99813 

0.99831 

0.99847 

0.99863 

3 

87 

0.99863 

0.99878 

0.99892 

0.99905 

0.99917 

0.99929 

o.99939 

2 

88 

0.99939 

0,99949 

0.99958 

0.99966 

0.99973 

0.99979 

0.99985 

I 

89 

0.99985 

0.99989 

o.99993 

0.99996 

0.99998 

1 . 00000 

r . 00000 

o° 

60'     50'    40'     30'     20'     10'     0'  A 

ogle 

COSJLNE 


224         TABLE  IV.— TRIGONOMETRIC  FUNCTIONS 

Natural  Tangents 


TANGENT 


Angle 

o'      io'     20'     30'     40'     50'     6o' 

89 

0° 

0.00000 

0.00291 

0.00582 

0.00873 

0.01164 

0.01455 

0.01746 

i 

0.01746 

0.02036 

0.02328 

0.02619 

0.02910 

0.03201 

0.03492 

88 

a 

0.03492 

0.03783 

0.04075 

0.04366 

0.04658 

0.04949 

0.05241 

87 

3 

0.05241 

0.05533 

0.05S24 

0.06116 

0.0640S 

0.06700 

0.06993 

86 

4 

0.06993 

0.07285 

0.0757SJ 

0.07870 

0.08163 

0.08456 

0.08749 

85° 

5° 

0.08749 

0.09042 

0.09335 

0.09629 

0.09923 

0. 10216 

0. 10510 

84 

6 

0. 10510 

0. 10805 

0. 11099 

0.11394 

0.1168S 

0.11983 

0. 12278 

83 

7 

0. 12278 

0.12574 

0. 12869 

0.13165 

0.13461 

O.I3758 

0. 14054 

82 

8 

0.14054 

0.1435* 

0. 14648 

O.I4945 

0.15243 

0.15540 

0.15838 

81 

9 

0.15838 

0.16137 

0.16435 

0.16734 

0.17033 

0.17333 

0.17633 

8o° 

10° 

o.i7633 

0.17933 

0.18233 

0.18534 

0.18835 

0.19136 

0.19438 

79 

ii 

0.19438 

0. 19740 

0. 20042 

0.20345 

0. 20648 

0.20952 

0.21256 

78 

12 

0.2125' 

0. 21560 

0. 21S64 

0. 22169 

0.22475 

0. 22781 

0.23087 

77 

13 

0.23087 

0.23393 

0. 23700 

0. 24008 

0.24316 

0.24624 

0.24933 

76 

14 

0.24933 

0. 25242 

0.25552 

0.25862 

0.26172 

0.26483 

0.26795 

75° 

15° 

0.26795 

0.27107 

0.27419 

0.27732 

0.28046 

0.28360 

0.28675 

74 

16 

0.28675 

0.28990 

0.29305 

0.29621 

0.29938 

0.30255 

o.30573 

73 

i7 

0.30573 

0.30S91 

0.31210 

0. 3*530 

0.31850 

0.32171 

0.32492 

72 

18 

0.32492 

0.32814 

0. 33*36 

0.33460 

0.33783 

0. 3410S 

o.34433 

71 

i9 

0.34433 

o.34758 

0.35085 

0.354*2 

o.35740 

0.36068 

0.36397 

70° 

20° 

o.36397 

0.36727 

0.37057 

0.37388 

0.37720 

0.38053 

0.38386 

69 

21 

0.38386 

0.38721 

0.39055 

o.3939i 

0.39727 

0.40065 

0.40403 

68 

22 

0.40403 

0.40741 

0.41081 

0.41421 

0.41763 

0.42105 

0.42447 

67 

23 

0.42447 

0.42791 

0. 43*36 

0.43481 

0.43828 

0.44I75 

0.44523 

66 

24 

o.44523 

0.44872 

0.45222 

0-45573 

0.45924 

0.46277 

0.46631 

6S° 

25° 

0.46631 

0.46985 

o.4734i 

0.47698 

0.48055 

0.48414 

0.48773 

64 

26 

0.48773 

0.49134 

0.49495 

0.49S5S 

0. 50222 

0.50587 

0.50953 

63 

27 

o.50953 

0.51320 

0.516S8 

0.52057 

0.52427 

0.52798 

0.53*7* 

62 

28 

0.53I71 

o.53545 

0.53920 

0.54296 

0.54673 

o.5505* 

o.5543i 

61 

29 

0.55431 

0.55812 

0.56194 

0.56577 

0.56962 

0.57348 

o-57735 

6o° 

30° 

o.57735 

0.58124 

0.58513 

0.58905 

0.59297 

0.59691 

0. 60086 

59 

31 

0.600S6 

0.604S3 

0.60881 

0.612S0 

0.61681 

0.62083 

0.62487 

58 

32 

0.62487 

0.C2S92 

0.63299 

0.63707 

0.64117 

0.64528 

0.64941 

57 

33 

3.64941 

0.65355 

0.65771 

0.66189 

0.66608 

0.67028 

0.6745* 

56 

34 

0.67451 

0.67S75 

0.6S301 

0.68728 

0.69157 

0.695S8 

0.70021 

55° 

35° 

0.  70021 

0.70455 

0.70S91 

0.7*329 

0.71769 

0. 72211 

0.72654 

54 

36 

0.72654 

0. 73100 

0.73547 

0.73996 

0.74447 

0.74900 

0.75355 

53 

37 

o.75355 

0.75S12 

0.76272 

0.76733 

0.7719G 

0. 77661 

0. 78129 

52 

38 

0.78129 

0.78598 

0. 79070 

o.79544 

0.80020 

0. 80498 

0.80978 

5* 

39 

0.80978 

0.81461 

0.81946 

0.S2434 

0.82923 

0.83415 

0.83910 

50° 

40° 

0.83910 

0.84407 

0.  34906 

0.85408 

0.85912 

0.86419 

0.86929 

49 

41 

0.86929 

0.S7441 

0.87955 

0.88473 

0.88992 

0.89515 

0.90040 

48 

42 

0.90040 

0.90569 

0.91099 

0.91633 

0. 92170 

0.92709 

0.93252 

47 

43 

0.93252 

0.93797 

0.94345 

0.94896 

0-9545* 

0.96008 

0.96569 

46 

44 

0.96569 

0. 97*33 

0.97700 

0.98270 

0.98843 

0.99420 

1 . 00000 

45° 

60'    50'     40'    30'     20'     10'     o'  / 

ingle 

COIANGENX 


TABLE  IV.- 
and  Cotangents 


-TRIGONOMETRIC  FUNCTIONS         225 


TANGENT 


Angle 

0'     io'     20'    30'     4°'     50'    60' 

45°  1 

1. 00c 00 

1.00583 

1.01170 

1.01761 

1-02355 

1.02952 

1-03553 

44 

46 

1-03553 

1. 04158 

1.04766 

i-°5378| 

1.05994 

1. 06613 

1.07237 

43 

47 

1.07237 

1.07864 

1.08496 

I-09I31! 

1.09770 

1 . 10414 

1 . 11061 

42 

48 

1. 11061 

1.11713 

1. 12369 

1. 13029 

1. 13694 

1. 14363 

1-15037 

41 

49 

I.I5037 

I.IS7I5 

1. 16398 

1. 17085 

i-J7777 

1. 18474 

I.I9I75 

40° 

50° 

I.I9I7S 

1. 19882 

1-20593 

1. 21310 

1. 22031 

1.22758 

1.23490 

39 

5i 

1.23490 

1.24227 

1.24969 

1. 25717 

1. 26471 

1.27230 

1.27994 

38 

52 

1.27994 

1.28764 

1. 29541 

1.30323 

1.31110 

1. 31904 

1.32704 

37 

53 

1.32704 

1.335" 

1.34323 

I.35M2 

L35968 

1.36800 

1-37638 

36 

54 

1.37638 

1.38484 

I-39336 

1. 40195 

1.41061 

I.4I934 

1. 42815 

35° 

55° 

1. 42815 

L43703 

1.44598 

I.45501 

1. 4641 1 

1-47330 

1.48256 

34 

56 

1.48256 

1. 49190 

i-5OI33 

1. 51084 

i.52°43 

1. 53010 

I-53987 

33 

57 

1.53987 

I-54972 

1.55966 

1.56969 

1. 57981 

1.59002 

1.60033 

32 

58 

1.60033 

1. 61074 

1. 62125 

1. 63185 

1.64256 

1.65337 

1.66428 

31 

59 

1.66428 

I.67530 

1.68643 

1.69766 

1 . 70901 

1. 72047 

1-73205 

300 

60  ° 

1.73205 

1-74375 

1.75556 

1.76749 

1-77955 

1. 79174 

1 . 80405 

29 

61 

1 . 80405 

1. 81649 

1.82906 

1. 84177 

1.85462 

1.86760 

1.88073 

28 

62 

1.88073 

1 . 89400 

1. 90741 

1.92098 

1-9347° 

1.94858 

1. 96261 

27 

63 

1. 96261 

1.97680 

1.99116 

2.00569 

2.02039 

2.03526 

2.05030 

26 

64 

2.05030 

2.06553 

2.08094 

2.09654 

2.11233 

2. 12832 

2.14451 

25° 

65° 

2.14451 

2. 16090 

2.17749 

2.19430 

2. 21132 

2. 22857 

2. 24604 

24 

66 

2. 24604 

2.26374 

2.28167 

2. 299S4 

2.31826 

2.33693 

2-35585 

23 

67 

2.35585 

2.37504 

2-39449 

2.41423 

2.43422 

2.45451 

2.47509 

22 

68 

2.47509 

2-49597 

2-5I7I5 

2.53C65 

2. 56046 

2. 5S261 

2. 60509 

21 

69 

2.60509 

2.62791 

2.65109 

2.67462 

2.69853 

2. 72281 

2.74748 

20° 

70° 

2.74748 

2.77254 

2. 79802 

2.82391 

2.85023 

2.87700 

2. 90421 

19 

71 

2.90421 

2.93189 

2.96004 

2.98S69 

3.01783 

3-04749 

3-0776S 

18 

72 

3.07768 

3.10842 

3-I3972 

3-I7I59 

3. 20406 

3-237I4 

3.27085 

17 

73 

3-27oSs 

3-30521 

3-34023 

3-37594 

3-41236 

3-44951 

3-48741 

16 

74 

3-48741 

3.52609 

3.56557 

3.60588 

3-64705 

3.68909 

3-73205 

15° 

75° 

3.73205 

3-77595 

3.82083 

3.86671 

3-9J364 

3.96165 

4.01078 

14 

76 

4.01078 

4.06107 

4. 11256 

4-16530 

4.21933 

4.27471 

4-33J48 

13 

77 

4.33148 

4-38969 

4-44942 

4-5IQ7i 

4.57363 

4.63825 

4.70463 

12 

78 

4-70463 

4. 77286 

4.84300 

4-9I5l6 

4.98940 

5.06584 

5-14455 

11 

79 

5-14455 

5.22566 

5.30928 

5-39552 

5- 4845 J 

5.57638 

5.67128 

10° 

8o° 

5.67128 

5-76937 

5.87080 

5-97576 

6.08444 

6.19703 

6.31375 

9 

81 

6.313/5 

6.43484 

6.56055 

6.69116 

6. 82694 

6.96823 

7-"S37 

8 

82 

7- "537 

7.26873 

7.42871 

7-59575 

7.77035 

7-95302 

8.14435 

7 

83 

8.14435 

8.34496 

8-55555 

8.77689 

9.00983 

9.25530 

9.5!436 

6 

84 

9-5I436 

9.78817 

10.0780 

10.3854 

10.7119 

11.0594 

n.4301 

5° 

85° 

n.  4301 

11.8262 

12.2505 

12.7062 

13.1969 

13.7267 

14-3007 

4 

86 

14.3007 

14.9244 

15.6048 

16.3499 

17.1693 

18.0750 

19.0811 

3 

87 

19.081 1 

20. 2056 

21.4704 

22.9038 

24.5418 

26.4316 

28.6363 

2 

88 

28.6363 

31.2416 

34.3678 

38.1885 

42.9641 

49.1039 

57.2900 

I 

69 

57.2900 

68.7501 

8S-9398 

114-589 

171-885 

343-774 

00 

0° 

60'     50'     40'      30'     20'      10'     0'  J 

^ngle 

COTANGENT 


INDEX 


Abscissa,  114 
Addition  (algebraic),  50 

—  of  vectors,  183 
Angle,  119 

— ■  complementary,  120 

—  from  three  sides  given,  174 

-  phase,  180 

-  right,  119 

-  straight,  119 

—  supplementary,  120 

—  theorems,  120 
Antilogarithm,  137 
Area  of  a  triangle,  175 
Armature  winding,  46 
Axis,  110,  113 

Base,  29 

Characteristic,  132 

Checking  an  equation,  3 

Circle,  26,  125 

Circular  mil,  40 

Coefficient,  30 

Cologarithm,  140 

Composition  of  forces,  184 

Complementary  angles,  120 

Condenser,  37,  93 

Conductor  in  magnetic  field,  154 

Congruent  triangles,  121 


Coordinates,  113 
Cosecant,  150 
Cosine,  150 

—  law  of,  172 
Cotangent,  150 
Current,  alternating,  188 
Cutting  speed,  79 

Degree,  120 
Depression,  161 
Direct  proportion,  76 
Division,  64 

—  by  slide  rule,  18 

Efficiency,  71 

Electrical    applications    of    trigo- 
nometry, 177 
Elevation,  161 
Equation,  1 
— -  checking  of,  3 

—  fractional,  5 

—  graph  of,  115 

—  quadratic,  95 

—  simultaneous,  104 
Evaluation,  29,  31 

—  of  formulas,  22,  32 
Exponent,  31 

—  in  division,  65 

—  in  multiplication,  59 


227 


228 


INDEX 


Exponent,  unknown,  14.") 
Extremes,  71 

Field  intensity,  43 
Flux,  45 

Force  on  a  magnet,  38,  87 
Formulas,  32 

—  electrical,  33,  86 

—  evaluation  of,  22,  '■'>- 

—  quadratic  equation,  100 

—  square  and  square  root,  25,  86 
, —  transposition  of,  37 

Fractional  equations,  5 
Fractions     reduced     to     a     given 

denominator,  75 
Fulcrum,  56 
Functions,  150 

—  as  lines,  164 

—  logs  of,  160 

—  of  angles  greater  than  90°,  1(55 

( tears,  79,  83 
General  number,  29 
Geometry,  119 
Graph,  109 

—  of  an  equation,  115 

—  of  simultaneous  equations,  117 
Grouping,  31 

Heat  in  electric  current,  30 
Horse  power,  88 

Impedance,  192 
Inductance  coil,  92 
Instantaneous  voltage,  179 
Interpolation,  135,  158 
Inverse  proportion,  76 
Isosceles  triangle,  122 


Leverage,  56 

—  law  of,  60 
Lever  arm,  56 
Lines  of  force,  179 
Location  of  p<  ii  ts,  114 
Logarithms,  131 

Magnetic  circuit,  45 
Mantissa,  132 
Means,  74 
Member,  7 
Mil,  40 
Minute,  120 
Monomial,  60 
-  division  of,  59 

—  multiplication  of,  65 
Multiplication,  "<' 

—  by  slide  rule,  19 
Murray  loop.  91 

Negative  numbers,  49 
Numbers,  negative,  49 
positive,  49 

Oblique  triangle,  169 
Ohm's  law,  34 
Ordinate,  114 
Origin,  114 

Parallel  lines,  123 
Parallelogram,  125 
Parenthesis,  31 
Percentage,  69 
Perigon,  120 
Phase  angle,  180 
Polynomials,  62,  63 
Positive  numbers,  49 
Power,  30,  142 


INDEX 


229 


Pow  T  factor,  191 

-  in  A.  C.  circuit,  189 

—  in  D.  C.  circuit,  86 
Projection,  177 
Proportion,  23,  68,  72,  127 

direct,  76 

—  inverse,  76 
Protractor,  120 
Pulleys,  79 

—  speed  of,  82 

Quadratic  equation,  95 

—  by  completing  square,  98 

—  by  formula,  100 
Quadrilaterals,  124 

Ratio,  68,  127 

—  given,  71 
Rectangle,  125 
Resistances  in  parallel,  39 

—  in  series,  33 
Resultant,  184 
Rhombus,  125 
Right  angle,  119 

—  triangle,  101,  128 

—  triangulation,  148 
Rim  speed,  79 
Roots  of  numbers,  143 
Rotating  loop,  179 

Scalar  quantity,  182 

Secant,  150 

Second,  120 

Separating  in  a  given  ratio,  71 

Signs,  29 

Similar  triangles,  128 

Simultaneous  equations,.  104 

—  by   addition   and   subtraction, 

104 


Simultaneous    equation,    by    sub- 
stitution, 107 

—  graph  of,  117 
Sine,  150 

-  law  of,  169 
Slide  rule,  13,  131 

—  division  by,  18 

—  general  suggestions  for,  26 

—  locating  decimal  point,  20 

—  multiplication  by,  19 

—  scales,  14,  24 

—  types  of,  13 

—  use  in  special  computations,  22, 

23,  25,  26,  27 
Square,  125 
Square  and  square  root  formulas, 

86 
Straight  angle,  119 
Subtraction  (algebraic),  .50 

—  of  vectors,  185 
Supplementary  angles,  120 
Surface  speed,  79 

Tangent,  150 

—  law  of,  174 

Temperature  change  of  resistance, 

42 
Term,  7 

Transformation  of  formulas,  37 
Transposition,  7 
Trapezium,  124 
Trapezoid,  124 
Triangle,  120 

—  area  of,  175 
— -  congruent,  121 

—  isosceles,  122 

—  oblique,  169 

—  right,  101,  128 

—  similar,  128 


230 


INDEX 


Triangulation,  right,  148 
Trinomial  square,  97 
Turning  moment,  56 

Unknown  exponent,  145 

Vectors,  182 

—  difference  of,  185 


Vectors,  sum  of,  18.3 

Vertex,  119 

Voltage  of  D.  C.  generator,  90 

Wheatstone  bridge,  71 
Work  done  by  an  electric  current, 
35 


UN,VERS1TY  OF  CAL.FOKN.A  LIBRARY 

Los  Angeles 
ThisbooKisC,ueon,he1a.,-a.e...mp.— • 


University  of  California.  Los  Anaeles 


L  005  839  144  2 


AA    000  790  291    9 


SC  RN  BRANCH, 

UNIVERSITY  OF  CALIFORNIA, 

LLBRARY, 

'LOS  ANGELES,  CALIF. 


